This paper uses exponential sum methods to show that if \(E \subset \mathcal (\mathbb{Z}/p^r)^n \setminus (p)^{(n)}\) has a sufficiently large density and \(j\) is any unit in the finite ring \(\mathbb{Z}/p^r\) then there exist pairs of elements of \(E\) whose dot product equals \(j\). It then applies this to the problem of detecting \(2-\) simplices with endpoints in \(E\).
This article studies the distribution of dot product values over \({\mathbb Z}_q := {\mathbb Z}/ q\) on sets \(E \times E \subset {\mathbb Z}_q^{2 n}\) (\(n \ge 2\)) which belong to the unit group \(U_q := ({\mathbb Z}/q)^\times\), where \(q = p^r \ \ \mbox{ and $r \ge 2$}.\)
If \(n \ge 2\) our main result (Theorem 1.1) finds a simple lower bound on the density \(\delta_E := |E|/q^n\) of a subset \(E \subset \mathcal {\mathbb Z}_q^n \setminus (p)^{(n)}\), where \[(p)^{(n)} := \{(a_1, \dots, a_n) \in {\mathbb Z}_q^n : \ \ a_i \in (p) \ \ \mbox{ for each $i$}\}\,,\] which is uniformly bounded from below in all \(p \gg 1\) and all \(r \ge 2\), and which insures that the following property is satisfied:
for each \(j \in U_q\) there exist \[{\bf x}= (x_1,\dots, x_n), \ {\bf y}= (y_1,\dots, y_n)\in E \text{ s.t. }\langle {\bf x}, {\bf y}\rangle := \sum_{i=1}^n x_i y_i = j \in {\mathbb Z}_q.\]
Defining for each \(j \in U_q\) and \(E \subset \mathcal {\mathbb Z}_q^n \setminus (p)^{(n)}\) \[\beta_j (E) := \big|\{({\bf x}, {\bf y}) \in E^2 : \langle {\bf x}, {\bf y}\rangle = j\}\big|\,,\] the first observation to make is that the restriction to subsets \(E\) of \({\mathbb Z}_q^n \setminus (p)^{(n)}\) is a reasonable one since any subset of \(E\) that lies inside \((p)^{(n)}\) cannot contribute to \(\beta_j (E)\) because \(j \notin (p).\)
What we mean precisely by the expression “\(\delta_E\) is uniformly bounded from below in all \(p \gg 1\) and \(r \ge 2\)” is that there exist \(\gamma < 0\), \(P \ge 1\), and \(C > 0\) (independent of \(p, r\)) such that
Our main result shows this property holds if \(\gamma = -(n-1)/2\), and also gives an explicit main term for each \(\beta_j (E)\). Its proof is given in Section 2.2.4 and is a simple consequence of the two estimates derived in Section 2.2.2, Section 2.2.3. The statement is as follows.
Theorem 1.1. Assume \(n \ge 2\) and \(E \subset {\mathbb Z}_q^n \setminus (p)^{(n)}.\) Then there exists a constant \(C > 0\) (uniform in \(p \gg 1\) and \(r \ge 2\)) such that \[\label{dot product count} p \gg 1 \ \mbox{ and } \ \delta_E \ge\ C \, p^{-\frac {n-1}2} \ \ \mbox{ implies} \ \ \beta_j (E) = \frac{|E|^2}q \cdot (1 + o(1)) \quad \mbox{ for each $j \in U_q$ and $r \ge 2$}. \tag{1}\]
The significance of Theorem 1.1 is that our density lower bound therefore only depends upon a sufficiently large \(p\) and not upon a choice of \(r \ge 2\). This is to be contrasted with the results in [1], [8] in which the weaker density lower bound \(\delta_E \gg \, r \, p^{- \frac {n-1}2}\) sufficed to insure that \(\beta_j (E)\) is positive when \(r \ge 1\) (for each \(j\) and \(n \ge 2\)), but without giving the main term in (1).
An additional point to emphasize here is that our density lower bound is on a scale of \(p\) and not \(q\), i.e. it is a negative power of \(p\) not \(q.\) This appears to be an inevitable consequence of the presence of zero divisors in \({\mathbb Z}_q^n\), that is, vectors that are zero after multiplication by some power of \(p\) less than \(r.\)
Since our method relies upon estimates for averages of exponential sums mod \(q\) of scalar products of non zero vectors, such averages are necessarily inflated because of this phenomenon (see (23), (24)ff. in Section 2.2.2). However, by being careful and using a different approach to that in [1], [8], we are able to extract some savings that result in the negative exponent of \(p\) in (1).
A variant of Theorem 1.1 is the subject in Section 3. This concerns the problem of detecting points in a “\(2-\)simplex”, which for our purposes here means any set of the form \[\mathbf \Sigma_{{\bf v}, j} (E) = \{({\bf x}_1, {\bf x}_2) \in E \times E : \|{\bf x}_1\| = v_1\,, \ \|{\bf x}_2\| = v_2\,, \ \langle {\bf x}_1, {\bf x}_2 \rangle = j\}\,,\] where \(({\bf v}, j) = (v_1, v_2, j) \in U_q^3\) and \(\|{\bf x}\| = \langle {\bf x}, {\bf x}\rangle\,.\)
It is not difficult to adapt the argument that proves Theorem 1.1 to this problem. It suffices to replace \(E\) by \(E_{ v_1} \times E_{v_2}\) where \[E_{v_u} := E \cap S_{v_u} := E \cap \{{\bf x}: \|{\bf x}\| = v_u\} \qquad (u = 1, 2).\]
However, an apparent obstacle exists that prevents our method from being applicable for a given \(n \ge 2\) and all \(r \ge 2\).
To see this most simply, assume that \(E\) is a random subset of \({\mathbb Z}_q^n\), which tells us that \(E\) and the \(S_{\nu_u}\) are independent sets with respect to the probability measure \[X \subset {\mathbb Z}_q^n \longrightarrow \delta_X.\] In this event it follows that for each \(u\): \[\delta_{E_{v_u}} = \delta_E \, q^{-1} \, (1 + o(1))\,.\]
By combining (35) and (34), we show (see Section 3) that \[\delta_{E} \, q^{-1} \, (1 + o(1)) \gg q^{-\frac 12} \, p^{-\frac {n-2}2} \ \ \ \implies \ \ |\Sigma_{{\bf v}, j} (E)| = \frac{|E_{v_1}| \, |E_{v_2}|}q \, (1 + o(1)) = \frac{|E|^2}{q^3} \, (1 + o(1)).\] However, since \(\delta_E \le 1\), the hypothesis can only be satisfied when \(r < n – 2\) and \(p \gg 1.\)
More generally, our second result includes the possibility that \(E\) and the \(S_v\) are not independent.
Theorem 1.2. Assume \(n \ge 5\) and \(r \ge 2.\) Then there exists a constant \(c> 0\) (uniform in \(p \gg 1\), \(r \ge 2\)) such that for all \(({\bf v}, j) \in U_q^3\) : \[\label{condition 1*} p \gg 1, r < n – 2, \ \mbox{ and } \ \delta_{E_{v_1}}^{\frac 12}\,\, \delta_{E_{v_2}}^{\frac 12} \ge c \,\, q^{- \frac 12} \, p^{-\frac {(n-2)}2} \ \ \mbox{ implies } \ \ |\,\mathbf \Sigma_{{\bf v}, j} (E)\,| = \frac {|E_{v_1}| \, |E_{v_2}|}q \, (1 + o(1)). \ \tag{2}\]
In general, it is convenient to introduce a parameter \(\alpha_u\), defined by setting \[\delta_{E_{\nu_u}} := \alpha_u \, \delta_E \, \delta_{S_{\nu_u}}, \tag{3}\] and express the criterion of Theorem 1.2 in terms of \(\alpha_1, \alpha_2\), and \(|E|\) as follows. \[\label{condition 1**} p \gg 1, \ r < n – 2, \ \ \mbox{ and } \ \ \delta_E \ge c \,\, \frac {q^{\frac 12} \, p^{-\frac {(n-2)}2}}{(\alpha_1 \, \alpha_2)^{\frac 12}} \quad \mbox{ implies } \quad |\,\mathbf \Sigma_{{\bf v}, j} (E)\,| = \frac {\alpha_1 \, \alpha_2\, |E|^2}{q^3} \, (1 + o(1)). \tag{3}\]
What this means in practice is that the parameters \(\alpha_1, \alpha_2\) are implicitly constrained by the condition that any lower bound for \(\delta_E\) must be at most \(1\), that is, \[\alpha_1 \, \alpha_2 \gg p^{r – (n-2)}.\]
An immediate corollary of Theorem 1.2 extends the main result in [4] to detect “circular \(2-\)point configurations” whose vertices belong to some \(E \subset{\mathbb Z}_q^n\) for all \(p \gg 1\) and \(n \ge 5\), provided that \(2 \le r < n – 2\).
Given a set \(E \subset {\mathbb Z}_q^n\) and unit \(\iota\) this problem asks whether pairs of points \(({\bf x}_1, {\bf x}_2) \in E^2\) exist that satisfy simultaneously the conditions: \[\|{\bf x}_1\| = \|{\bf x}_2\| = \iota \quad \mbox{ and } \quad \|{\bf x}_1 – {\bf x}_2\| = \iota.\]
Given that \(2 \cdot \iota \neq \iota \, mod\, p\) when \(p > 2\), such a property follows trivially from the hypotheses in (2) by setting \(v_1 = v_2 = \iota\) and choosing \(j = 2^{-1} \cdot \iota\), which insures that \(\iota = 2 \, (\iota – j)\,.\) Since there are \(|U_q|\) possible \(\iota\) there are \(|U_q|\) such circular \(2-\)point configurations if the hypotheses in (2) hold.
Theorem 1.2 can also be interpreted in terms of the well known graph coloring problem from geometric Ramsey theory in the simplest possible case. In this context, the value of \(\|{\bf x}_1 – {\bf x}_2\|\) becomes a color for the edge of a graph whose vertex set \(\{{\bf x}_1, {\bf x}_2\}\) is a subset of some \(E \subset {\mathbb Z}_q^n\). Theorem 1.2 tells us that if \(p\) sufficiently large then \(|U_q|\) distinct monocolorings of the edge set must exist provided that \(E\) has density at least \(q^{\frac 12} \, p^{- \frac{n-2}2}\) and \(r < n – 2\).
It is natural to ask whether the proof of Theorem 1.2 can be extended to address analogous problems for complete graphs in \({\mathbb Z}_q^n\) with \(k \ge 3\) vertices. Using very different ideas, some progress on this problem is given in [6].
Theorems 1.1 and 1.2 (as well as the theorems proved in [4]-[5]) can be thought of as a modest response to Tao’s general challenge (see [7]) to extend to finite rings results in additive combinatorics involving orthogonal invariants that have been proved over finite fields, using ideas or techniques that do not typically (or immediately, or even at all) extend to finite \(p-\)adic rings. Although Tao’s explicit interest in [ibid.] concerned the sum-product phenomenon, it does not seem unreasonable to adopt a broader understanding of his challenge to include, in addition, point configuration questions like those addressed in [op.cit.] and this article.
A basic feature of Theorem 1.1 is that it proves a result that is uniform in \(r \ge 2\), provided only that \(p\) is larger than a lower bound that does not depend upon \(r\). In addition, it will be evident that there is nothing in the proof of Theorem 1.1 that would not extend straightforwardly to treat the same problem over finite subrings of the ring of integers of any finite extension of the \(p-\)adic field \({\mathbb Q}_p.\) Filling in the details of such a discussion is an exercise best left to the reader. On the other hand, because the density lower bound in Theorem 1.1 is a power of \(p\), and not \(q\), it does not seem possible to improve Theorem 1.2 by eliminating the a priori bound for \(r\).
In a Concluding Remark we indicate a simple application of the uniformity in \(r\) property of Theorem 1.1 to a comparable dot product problem over the ring \({\mathbb Z}_p\) of \(p-\)adic integers. An interesting consequence of our result is that by using only Haar measure we can say something about the presence of dot product values created by points in a subset \(E \times E\) when \(E\) is a closed subset of \({\mathbb Z}_p^n\). This is sketched at the end of the article. An alternative approach, for a general configuration problem on average, was worked out in the thesis [2], which used the full arsenal of geometric measure theoretic techniques based upon Hausdorff and Frostman measures adapted to the non archimedean metric context.
We use throughout the particular notations below where \(j \in U_q\), \({\bf m}= (m_1,\dots, m_n) \in {\mathbb Z}_q^n\), \(\pmb m = ({\bf m}_1, {\bf m}_2) \in {\mathbb Z}_q^{2 n}\), and \({\bf x}= (x_1,\dots, x_n), \ {\bf y}= (y_1, \dots, y_n) \in {\mathbb Z}_q^n\) : \[\begin{aligned} & \langle {\bf x}, {\bf y}\rangle = \sum_{i=1}^n x_i y_i\,; \quad \langle {\bf x}, {\bf m}\rangle := \sum_i x_i m_i \,; \ \|{\bf x}\| := \sum_{i=1}^n x_i^2 \ \ (\mbox{ all defined in ${\mathbb Z}_q$}); \\ & ord_p \,{\bf x}= min_i\, ord_p \, x_i := min_i \ \mbox{max\,$\{\ell : p^\ell | x_i \}$}; \\ &1_j ({\bf x}, {\bf y}) = \mbox{ characteristic function of } \ \ \{({\bf x}, {\bf y}) \in {\mathbb Z}_q^{2 n} : \langle {\bf x}, {\bf y}\rangle = j\}\,; \\ &1_E ({\bf x}) = \mbox{characteristic function of $E$}\,;\\ & \mbox{ for any $y \in {\mathbb Z}_q$} \ \ \chi (y) := e^{2 \pi i y/q}\,; \nonumber \\ &\widehat 1_j (\pmb m) = q^{- 2 n} \ \sum_{{\bf x}\in {\mathbb Z}_q^n} \ 1_j ({\bf x}, {\bf y}) \chi \langle ({\bf x}, {\bf y}), -\pmb m \rangle\,; \quad \widehat 1_E (\pmb m) = q^{-n} \sum_{{\bf x}\in {\mathbb Z}_q^n} \ 1_E ({\bf x}) \chi \langle {\bf x}, -{\bf m}\rangle\,; \\ &\mbox{ for any finite set $X$,} \quad |X| = \mbox{ number of elements of $X$}\,;\nonumber \\ &c_p = 1 – p^{-1}\,, \ \ c_{p, n} = 1 – p^{-r n}\,, \ \ C_{p, r, n} = c_p \, c_{p, n} \, \frac {1 – p^{- r (n – 1)}}{1-p^{-(n – 1)}},\\ &\delta_E = |E|/q^n \,. \end{aligned}\]
It is clear that \[\begin{aligned} \beta_j (E) =& \sum_{\pmb x = ({\bf x}_1, {\bf x}_2) \in {\mathbb Z}_q^{2 n}} \ 1_E ({\bf x}_1) \cdot 1_E ({\bf x}_2) \cdot 1_j (\pmb x) \nonumber \\ =& \sum_{\pmb m = ({\bf m}_1, {\bf m}_2)} \widehat 1_j (\pmb m) \ \sum_{\pmb x = ({\bf x}_1, {\bf x}_2) \in {\mathbb Z}_q^{2 n}} \ 1_E ({\bf x}_1) \cdot 1_E ({\bf x}_2) \cdot \chi \langle \pmb x, \pmb m \rangle \nonumber \\ =& q^{2 n} \cdot \sum_{\pmb m = ({\bf m}_1, {\bf m}_2)} \widehat 1_j (\pmb m) \cdot \widehat 1_E (-{\bf m}_1) \cdot \widehat 1_E (-{\bf m}_2) := \mathcal M^* + \mathcal E^* , \nonumber \end{aligned}\] where \[\begin{aligned} \label{error terms} \mathcal M^* :=& q^{2 n} \cdot \widehat 1_j ( 0, 0) \cdot \widehat 1_E ( 0)^2 \nonumber \\ \mathcal E^* :=& q^{2 n} \cdot \sum_{\pmb m \neq ( 0, 0)} \widehat 1_j (\pmb m) \cdot \widehat 1_E (-{\bf m}_1) \cdot \widehat 1_E (-{\bf m}_2) \nonumber \\ =& q^{2 n} \cdot \bigg\{\widehat 1_E ( 0) \cdot \sum_{{\bf m}_2 \neq 0} \widehat 1_j ( 0, {\bf m}_2) \cdot \widehat 1_E (-{\bf m}_2) + \sum_{{\bf m}_1 \neq 0} \sum_{{\bf m}_2} \widehat 1_j (\pmb m) \cdot \widehat 1_E (-{\bf m}_1) \cdot \widehat 1_E (-{\bf m}_2)\bigg\} \nonumber \\ :=& q^{2 n} \cdot \bigg[\delta_E \cdot \sum_{{\bf m}_2 \neq 0} I ({\bf m}_2) + \sum_{{\bf m}_1 \neq 0} \sum_{{\bf m}_2} II (\pmb m)\bigg]\,. \nonumber \\ :=& \mathcal E^*_I + \mathcal E^*_{II}. \end{aligned} \tag{4}\]
We think of \(\mathcal M^*\) as an expected main term and \(\mathcal E^*\) as an expected error term that must be shown to be strictly smaller than \(\mathcal M^*\) provided only that \(p\) and \(\delta_E\) are sufficiently large in the sense given in the Introduction.
We first evaluate \(\mathcal M^*\) when \(p \neq 2.\) It is clear that \[\widehat 1_E ( 0) = \frac {|E|}{q^n} = \delta_E.\]
To evaluate \(\widehat 1_j ( 0, 0)\) we note that the choice of the \(\pmb x = ({\bf x}_1, {\bf x}_2)\) coordinates on \({\mathbb Z}_q^{2 n}\) is not optimal to do this since it obscures the relation to the work in [4]. It is more useful first to define \[{\bf z}_1 = {\bf x}_1 + {\bf x}_2 \qquad {\bf z}_2 = {\bf x}_1 – {\bf x}_2\,.\]
Setting \(\pmb z = ({\bf z}_1, {\bf z}_2)\), since \(p \neq 2,\) it follows that \[\label{a coordinate change} T (\pmb z) = (T_1 (\pmb z), T_2 (\pmb z)) = \frac 12 \cdot \left({\bf z}_1 + {\bf z}_2, {\bf z}_1 – {\bf z}_2 \right) = ({\bf x}_1, {\bf x}_2) , \tag{5}\] defines a coordinate change on all of \({\mathbb Z}_q^{2 n}\) such that \[Q : \pmb z \longrightarrow \langle ({\bf x}_1 \circ T) (\pmb z), ({\bf x}_2 \circ T) (\pmb z) \rangle = \frac 14 \cdot \left(\|{\bf z}_1\| – \|{\bf z}_2\| \right)\,.\]
It is then clear that \(1_j^* (\pmb z) := 1_j \circ T (\pmb z)\) denotes the characteristic function of the level set \(\{{\bf z}: Q ({\bf z}) = j\}\). Since \(Q\) is an additive and nondegenerate quadratic form in \(2 n\) variables we deduce from [9] and a standard \(p-\)adic lifting argument, using Hensel’s Lemma, which lifts solutions of the congruence \(Q \equiv j \, mod \, p^i\) (\(i = 1, 2, \dots, r-1\)) to solutions of \(Q \equiv j \, mod \, p^{i+1}\), that \[\label{Qj count} \widehat 1_j ( 0, 0) = \widehat 1_j^* ( 0, 0) = q^{-1} \cdot \left(1 + O\left(p^{-\frac {2 n – 1}2}\right)\right) \quad \mbox{ uniformly in $r \ge 1$ and $j \in U_q$}\,. \tag{6}\]
We conclude \[\label{main term} \mathcal M^* = \frac {|E|^2}q \cdot \left(1 + O(p^{-\frac{2 n – 1}2})\right). \tag{7}\]
To simplify the discussion below, it will be useful here to fix an integer \(P_0\) such that \[\label{constant value} p \ge P_0 \implies \frac 12 \le 1 + O \left(p^{- \frac {2 n – 1}2}\right) \le 2. \tag{8}\]
It is convenient to split the work into two parts. We first bound \(\mathcal E_I^*\) (see (13)) whose proof only relies upon the fundamental exponential sum estimate of Weil that is used to prove (6). We then use a very different method to establish a different bound \(\mathcal E_I^*\) in Section 2.2.2 (see (35)). We have thought it instructive to include a discussion of each method in order to highlight the relative strength of the second approach.
Of significance here, however, is that is possible to adapt the second method to bound \(\mathcal E_{II}^*\) in Section 2.2.3. We finish the proof of Theorem 1.1 in Section 2.2.4 by showing that if \(p\) is sufficiently large (uniformly in \(r \ge 2\) and \(j \in U_q\)) and \(n \ge 2,\) then the expected main term \(\mathcal M^*\) is larger than \(\mathcal E^*\) provided that \(\delta_E\) is bounded below by the product of a constant with \(p^{-\frac{n-1}2}\) uniformly in \(j\), \(r \ge 2\), and all \(p \gg 1.\)
Remark 2.1. The reader might be curious about the case when \(r = 1,\) that is, when \({\mathbb Z}_q = {\mathbb F}_p.\) The method in [1] clearly suffices for this case. So there is no need to consider this possibility in the following discussion.
We first apply Cauchy-Schwarz and Plancherel’s Theorem to the sum over \({\bf m}_2 \neq 0\) which implies \[\begin{aligned} \label{CS 1} \mathcal E_I^* \le& q^n \cdot |E| \cdot \delta_E^{1/2} \cdot \left(\sum_{{\bf m}_2 \neq 0} |\widehat 1_j ( 0, {\bf m}_2)|^2 \right)^{1/2} \nonumber\\ =& q^n \cdot |E| \cdot \delta_E^{1/2} \cdot \bigg[\sum_{{\bf m}_2 \neq 0} \widehat 1_j ( 0, {\bf m}_2) \cdot \widehat 1_j ( 0, -{\bf m}_2) \bigg]^{1/2}\,. \end{aligned} \tag{9}\]
Second, we note that the definition of each Fourier transform in (9) implies \[\begin{aligned} \label{FT 1} \sum_{{\bf m}_2 \neq 0} \widehat 1_j ( 0, {\bf m}_2) \cdot \widehat 1_j ( 0, -{\bf m}_2) = q^{-4 n} \cdot \sum_{{\bf m}_2 \neq 0} \bigg[\sum_{{\bf y}} \sigma_j ({\bf y}) \, \chi \langle {\bf y}, -{\bf m}_2 \rangle \bigg]\cdot \bigg[ \sum_{{\bf y}'} \sigma_j' ({\bf y}') \, \chi \langle {\bf y}', {\bf m}_2 \rangle \bigg] , \end{aligned} \tag{10}\] where \[\begin{aligned} \sigma_j ({\bf y}) := \sum_{{\bf x}} 1_j ({\bf x}, {\bf y}) \, \quad \mbox {and } \quad \sigma_j' ({\bf y}') = \sum_{{\bf x}'} 1_j ({\bf x}', {\bf y}'). \end{aligned}\]
As a result, since \(\sigma_j ({\bf y}) = \sigma_j' ({\bf y})\) and the function \[({\bf y}, {\bf y}') \longrightarrow \sum_{{\bf m}_2 \neq 0} \ \chi \langle {\bf y}' – {\bf y}, {\bf m}_2 \rangle \ \ \mbox{ equals } \ \ ({\bf y}, {\bf y}') \longrightarrow q^n \cdot 1_{\{{\bf y}= {\bf y}'\}} ({\bf y}, {\bf y}') – 1\,,\] it follows that \[\begin{aligned} \label{weakness} \sum_{{\bf m}_2 \neq 0} \widehat 1_j ( 0, {\bf m}_2) \cdot \widehat 1_j ( 0, -{\bf m}_2) &= q^{-4 n} \cdot \sum_{{\bf y}, {\bf y}'} \sigma_j ({\bf y}) \sigma_j' ({\bf y}') \left( \sum_{{\bf m}_2} \chi \langle {\bf y}' – {\bf y}, {\bf m}_2 \rangle – 1 \right) \nonumber \\ &= q^{-4 n} \cdot \bigg\{ q^n \cdot \sum_{{\bf y}} \sigma_j^2 ({\bf y}) – \left(\sum_{{\bf y}} \sigma_j ({\bf y})\right) \cdot \left(\sum_{{\bf y}'} \sigma_j' ({\bf y}')\right)\bigg\} \nonumber \\ &= q^{- 3 n} \cdot \sum_{{\bf y}} \sigma_j^2 ({\bf y}) – q^{- 4 n} \cdot \left(\sum_{{\bf x}, {\bf y}} 1_j ({\bf x}, {\bf y}) \right)^2\,. \end{aligned} \tag{11}\]
Since \(j \in U_q,\) the sums over \({\bf y}, {\bf y}'\) are actually concentrated on \({\mathbb Z}_q^n \setminus (p)^{(n)}\). As a result, for each fixed \({\bf y}, {\bf y}' \in {\mathbb Z}_q^n \setminus (p)^{(n)}\): \[\sigma_j ({\bf y}) = \sigma_j' ({\bf y}') = q^{n-1} \,,\] which implies \[q^{- 3 n} \cdot \sum_{{\bf y}} \sigma_j^2 ({\bf y}) \le q^{- 3 n} \cdot q^{2 (n -1)} \cdot \big|\{{\bf y}\in {\mathbb Z}_q^n \setminus (p)^{(n)}\big| = q^{-2} \, (1 – p^{-n}) < q^{- 2}\,.\]
Moreover, (6) implies \[\sum_{{\bf x}, {\bf y}} 1_j ({\bf x}, {\bf y}) = q^{2 n – 1} \cdot \left(1 + O(p^{-\frac {2 n – 1}2})\right) \quad \mbox{ uniformly in $r \ge 1$ and $j \in U_q$}.\]
Combining these estimates, and using the facts that the sum over \({\bf m}_2\) is nonnegative, and for \(p \gg 1\) \[\left(1 + O(p^{- \frac {2 n – 1}2})\right)^2 = 1 + O(p^{- \frac {2 n – 1}2})\,,\] we conclude that \[\label{EI* bound} 0 \le \sum_{{\bf m}_2 \neq 0} \widehat 1_j ( 0, {\bf m}_2) \cdot \widehat 1_j ( 0, -{\bf m}_2) < \big| q^{-2} – q^{-2} (1 + O(p^{- \frac{2 n – 1}2}) \big| = q^{-2} \cdot O(p^{- \frac{2 n – 1}2})\,.\]
As a result, (9) now implies that there exists \(P_I'\) and a constant \(c_I'\) uniform in \(r \ge 2\), \(j\), and \(p \ge P_I'\) such that \[\label{EI* bound bis} p \ge P_I' \implies \mathcal E_I^* \le c_I' \cdot q^n \cdot |E| \cdot \delta_E^{1/2} \cdot \left( q^{-2} \cdot p^{-n + \frac 12})\right)^{1/2} = c_I' \cdot p^{-\frac n2 + \frac 14} \cdot q^{\frac n2 – 1 } \cdot |E|^{\frac 32}\,. \tag{13}\]
We start again with (9) but use a different idea to bound the rightmost factor where, implicitly, \(r \ge 2\) is always assumed.
Setting \(\pmb z = ({\bf z}_1, {\bf z}_2)\) and \(\pmb z' = ({\bf z}_1', {\bf z}_2'),\) and using the definition, it is clear that for each \({\bf m}_2 \neq 0\) \[\begin{aligned} \label{CS 1 bis1} |\widehat 1_j ( 0, {\bf m}_2)|^2 =& q^{- 4 n} \cdot \left(\sum_{\pmb z} 1_j (\pmb z) \chi \langle {\bf z}_2, -{\bf m}_2 \rangle \right) \cdot \left(\sum_{\pmb z'} 1_j (\pmb z') \chi \langle {\bf z}_2', {\bf m}_2 \rangle \right) \nonumber \\ =& q^{- 4 n} \cdot \left(\sum_{\pmb z, \pmb z'} 1_j (\pmb z) \cdot 1_j (\pmb z') \cdot \chi \langle {\bf z}_2, -{\bf m}_2 \rangle \cdot \chi \langle {\bf z}_2', {\bf m}_2 \rangle \right)\,. \end{aligned} \tag{14}\]
In which case, by exchanging the order of summation with \({\bf m}_2\) we see that \[\label{CS 1 bis2} \sum_{{\bf m}_2 \neq 0} |\widehat 1_j ( 0, {\bf m}_2)|^2 = q^{- 4 n} \cdot \left(\sum_{\pmb z, \pmb z'} 1_j (\pmb z) \cdot 1_j (\pmb z') \cdot \sum_{{\bf m}_2 \neq 0} \chi \langle {\bf z}_2, -{\bf m}_2) \cdot \chi \langle {\bf z}_2', {\bf m}_2) \right) . \tag{15}\]
We first note that since \(j \in U_q\) it follows that \(\pmb z, \pmb z' \notin (p)^{(2 n)}.\) For each \({\bf z}_1\) resp. \({\bf z}_1' \notin (p)^{(n)}\) set \(\mathbf \zeta ({\bf z}_1)\) resp. \(\mathbf \zeta ({\bf z}_1')\) to denote a fixed vector, depending upon \({\bf z}_1\) resp. \({\bf z}_1'\), in \[\{{\bf z}_2 : 1_j ({\bf z}_1, {\bf z}_2) = 1\} \quad \mbox{ resp.} \quad \{{\bf z}_2' : 1_j ({\bf z}_1', {\bf z}_2') = 1\}\,,\] and set \[\mathcal K ({\bf z}_1) = \{{\bf z}_2 : \langle {\bf z}_1, {\bf z}_2 \rangle = 0 \} \quad \mbox {resp.} \quad \mathcal K ({\bf z}_1') = \{{\bf z}_2' : \langle {\bf z}_1', {\bf z}_2' \rangle = 0 \}.\]
It then follows that \[\begin{aligned} \label{CS 1 bis3} &rhs (15) \nonumber \\ &= \sum_{{\bf z}_1, {\bf z}_1'} \ \sum_{\substack{{\bf w}_1 \in \mathcal K ({\bf z}_1)\\ {\bf w}_1' \in \mathcal K({\bf z}_1')}} \ \sum_{{\bf m}_2 \neq 0} \chi \langle \mathbf \zeta ({\bf z}_1), -{\bf m}_2 \rangle \cdot \chi \langle \mathbf \zeta ({\bf z}_1'), {\bf m}_2 \rangle \cdot \chi \langle {\bf w}_1, -{\bf m}_2 \rangle \cdot \chi \langle {\bf w}_1', {\bf m}_2 \rangle \nonumber \\ &=\sum_{{\bf z}_1, {\bf z}_1'} \ \sum_{{\bf m}_2 \neq 0} \chi \langle \mathbf \zeta ({\bf z}_1), -{\bf m}_2 \rangle \cdot \chi \langle \mathbf \zeta ({\bf z}_1'), {\bf m}_2 \rangle\ \sum_{\substack{{\bf w}_1 \in \mathcal K ({\bf z}_1)\\ {\bf w}_1' \in \mathcal K({\bf z}_1')}} \chi \langle {\bf w}_1, -{\bf m}_2 \rangle \cdot \chi \langle {\bf w}_1', {\bf m}_2 \rangle\,. \end{aligned} \tag{16}\]
We now must understand the behavior of the functions \[\label{expsum v} {\bf v}\neq 0 \longrightarrow \sum_{{\bf w}\in \mathcal K ({\bf z})} \ \chi \langle {\bf w}, {\bf v}\rangle \quad \mbox{ when ${\bf z}\in \{{\bf z}_1, {\bf z}_1'\}$ \ and \ ${\bf v}\in \{\pm {\bf m}_2\}$.} \tag{17}\]
To do this we use the fact that if \({\bf w}\in \mathcal K ({\bf z}) \longrightarrow \chi \langle {\bf w}, {\bf v}\rangle\) is a trivial resp. nontrivial homomorphism of the additive group \(\mathcal K({\bf z})\) then \[\label{basic count} \mbox{ the exponential sum in (17) equals $q^{n-1}$ resp. 0.} \tag{18}\]
To decide which possibility can occur it suffices to reduce the \(2 \times n\) matrix \[\mathcal L:= \left(\begin{matrix} {\bf z}\\ {\bf v}\end{matrix} \right) = \left(\begin{matrix} z_1 & z_2 & \cdots & z_n \\ v_1 & v_2 & \cdots & v_n \end{matrix} \right) ,\] to row echelon form. By a permutation of indices we may assume \(z_1 \in U_q\), in which event \(\mathcal L\) is row equivalent to \[\mathcal L'= \left(\begin{matrix} z_1 & z_2 & \cdots & z_n \\ 0 & v_2 – v_1 \,z_1^{-1} \, z_2 & \cdots & v_n – v_1 \,z_1^{-1} \, z_n \end{matrix} \right) ,\] where the inverse in \({\mathbb Z}_q\) of \(z_1\) is denoted \(z_1^{-1}\), and \[\mathcal K ({\bf z}) = \left\{ \left( -\big[\sum_{j=2}^n \ w_j \cdot \,z_1^{-1} \, z_j \big], \,w_2, \dots, w_n \right) : (w_2, \dots, w_n) \in {\mathbb Z}_q^{n-1} \right\}\,.\]
So, if \(v_1 \neq 0\) and \(\langle {\bf z}\rangle := \{\tau \, {\bf z}: \tau \in {\mathbb Z}_q\}\) then \[\label{key point1} {\bf v}\in \langle {\bf z}\rangle \iff {\bf w}\in \mathcal K({\bf z}) \to \chi \langle {\bf w}, {\bf v}\rangle \ \ \mbox{ is identically $1$}. \tag{19}\]
Indeed, the condition is clearly sufficient. The fact that it is also necessary is seen as follows. If for each \(u \ge 2\) we choose \(w_u\) to be a non zero divisor and set \[{\bf w}_u := w_u \cdot (-\,z_1^{-1} \, z_u, 0, \dots, 0, 1, 0,\dots, 0) \in \mathcal K ({\bf z}) \quad \mbox{ (where the $u^{th}$ entry equals $1$),}\] then for each \(u \ge 2\) \[\begin{aligned} & \chi \langle {\bf w}_u, {\bf v}\rangle = 1 \iff 0 = \langle {\bf w}_u, {\bf v}\rangle = w_u \cdot \left(v_u – v_1 \,\,z_1^{-1} \, z_u\right) \\ &\implies z_1 \,{\bf v}= {v_1} \,{\bf z}\implies {\bf v}= (z_1^{-1} \,v_1) \,{\bf z}\,\in \langle {\bf z}\rangle\,. \end{aligned}\]
In addition, since \(ord_p \,{\bf z}= 0,\) we also observe that if \(\nu := ord_p \,{\bf v}\), then \[\label {nu order issue} z_1 \,{\bf v}= {v_1} \,{\bf z}\implies \nu = ord_p\, v_1 \ \ \mbox{ and } \ \ \hat {\bf v}:= \ p^{-\nu} {\bf v}= (z_1^{-1} \,p^{-\nu} \, v_1) \,{\bf z}\in \langle {\bf z}\rangle \cap \left({\mathbb Z}_q^n \setminus (p)^{(n)}\right). \tag{20}\]
Moreover, if \(v_1 = 0\) then it is equally clear that \({\bf v}\neq 0\) cannot belong to \(\langle {\bf z}\rangle\), and \[\label {key point2} {\bf v}\neq 0 \implies \sum_{{\bf w}\in \mathcal K({\bf z})} \chi \langle {\bf w}, {\bf v}\rangle = \sum_{\widetilde {\bf w}= (0, w_2, \dotsc, w_n) \in {\mathbb Z}_q^{n-1}} \chi \langle \widetilde {\bf w}, {\bf v}\rangle = 0\, , \tag{21}\] since, in this event, the map \({\bf w}\in \mathcal K ({\bf z}) \longrightarrow \chi \langle {\bf w}, {\bf v}\rangle = \chi \langle \widetilde {\bf w}, {\bf v}\rangle\) is a nontrivial homomorphism (i.e., not identically equal to \(1\)) on the additive group \(\mathcal K ({\bf z})\).
This tells us that the rightmost sum in (16) can only be nonzero when \({\bf m}_2 \in \langle {\bf z}_1 \rangle \cap \langle {\bf z}_1' \rangle,\) in which event, (18) then implies this product equals \(q^{2 (n-1)}.\) In other words : \[\label{interim formula} rhs (15) = q^{-2 n – 2} \sum_{{\bf m}_2 \neq 0} \ \sum_{{\bf z}_1, {\bf z}_1' \notin (p)^{(n)}} \chi \langle \mathbf \zeta ({\bf z}_1), -{\bf m}_2 \rangle \cdot \chi \langle \mathbf \zeta'({\bf z}_1'), {\bf m}_2 \rangle \cdot 1_{\langle {\bf z}_1 \rangle \cap \langle {\bf z}_1' \rangle} ({\bf m}_2). \tag{22}\]
We next split up the innermost sum into that part over which \({\bf z}_1 = {\bf z}_1'\) and \({\bf z}_1 \neq {\bf z}_1'\) and bound each part separately.
If \({\bf z}_1 = {\bf z}_1'\) then we can evidently choose \(\zeta({\bf z}_1) = \zeta' ({\bf z}_1')\) and write each \({\bf m}_2 = p^{\nu} \, \widehat {\bf m}\) where \(\widehat {\bf m}\in {\mathbb Z}_{p^{r-\nu}}^n\) and \(ord_p \, \widehat {\bf m}= 0\), in which case, by thinking of \({\mathbb Z}_{p^{r-\nu}}^n\) as being embedded in \({\mathbb Z}_q^n\) as the vectors each of whose coefficients of a power of \(p\) larger than \(r-\nu-1\) equals \( 0\), we have: \[\label{first sum} \sum_{{\bf m}_2 \neq 0} \ \sum_{{\bf z}_1= {\bf z}_1' \notin (p)^{(n)}} \chi \langle \mathbf \zeta ({\bf z}_1), -{\bf m}_2 \rangle \cdot \chi \langle \mathbf \zeta'({\bf z}_1'), {\bf m}_2 \rangle \cdot 1_{\langle {\bf z}_1 \rangle} ({\bf m}_2) = \sum_{\nu = 0}^{r-1} \sum_{\widehat {\bf m}} \ \sum_{{\bf z}\notin (p)^{(n)}} \ 1_{\langle {\bf z}\rangle} (p^\nu \, \widehat {\bf m}). \tag{23}\]
Denoting the units in \({\mathbb Z}_{p^{r-\nu}}\) by \(U_{r-\nu}\), we then note that \[\label {units mod } p^\nu \, \widehat {\bf m}\in \langle {\bf z}\rangle \iff \mbox{ there exists $\eta \in U_{r-\nu}$ \ such that \ } \widehat {\bf m}= \eta \, {\bf z}\ mod \,p^{r – \nu}. \tag{24}\]
However, from this congruence one cannot infer that any of the coefficients of \(p^{r-\nu}, p^{r – \nu + 1},\) \(\dots,\) \(p^{r-1}\) of the \(p-\)adic expansion of \({\bf z}\) must also equal \( 0.\) So, any such \({\bf z}\) is only determined uniquely mod \(p^{r-\nu}.\)
Setting \(\eta' = \eta^{-1} \in U_{r-\nu}\) it follows that this property is equivalent to \[{\bf z}= \eta' \,\widehat {\bf m}\ mod \, p^{r – \nu}\,.\]
Since for fixed \({\bf m}\), \(\eta'\), and a fixed representative \(\tilde {\bf z}_0\) mod \(p^{r – \nu}\) of \(\eta' \, \widehat {\bf m}\) in \({\mathbb Z}_q^n\) there exist \(p^{\nu n}\) other vectors \(\tilde {\bf z}\in {\mathbb Z}_q^n\) such that \[\{\tilde {\bf z}_0 + p^{r – \nu} \tilde {\bf z}\} = \big\{{\bf z}\notin (p)^{(n)} \, mod \,q : {\bf z}= \eta' \widehat {\bf m}\ mod \, p^{r – \nu}\big\}\,,\] it follows that if \({\bf z}_1' = {\bf z}_1\) then: \[\begin{aligned} \label {sum value} rhs (23) = \sum_{\nu = 0}^{r – 1} \ | U_{r – \nu}| \, p^{\nu n} \, c_{p, n} &= c_p \, c_{p, n} \, q \, \sum_{\nu = 0}^{r – 1} p^{\nu (n – 1)} \nonumber \\ &= c_p \, c_{p, n} \, q \, p^{(r – 1) (n-1)} \, (1 – p^{- r (n-1)}) \, (1 – p^{-(n-1)})^{-1} \nonumber \\ &= \big[c_p \, c_{p, n} (1 – p^{- r (n-1)}) \, (1 – p^{-(n-1)})^{-1}\big] \, q^n \, p^{-(n-1)} \nonumber \\ &=C_{p, r, n} \, q^n \, p^{-(n – 1)}\,. \end{aligned} \tag{25}\]
If \({\bf z}_1 \neq {\bf z}_1'\) it follows that if \(\nu = ord_p \, {\bf m}_2\) then there exist \(\eta_0 = \eta_0 (\nu), \mu_0 = \mu_0 (\nu) \in U_{r – \nu}\) such that (as equations in \({\mathbb Z}_q^n\)) \[\begin{aligned} \label{constraint 1} {\bf m}_2 \in \langle {\bf z}_1 \rangle \cap \langle {\bf z}_1' \rangle \implies& {\bf m}_2 = p^\nu \, \widehat {\bf m}:= (p^{\nu} \,\eta_0) \cdot {\bf z}_1 = (p^{\nu} \, \mu_0) \cdot {\bf z}_1' \ \nonumber\\ \implies& {\bf z}_1' = \eta_0 \,\mu_0^{-1} \, {\bf z}_1 \ mod \, p^{r – \nu}. \end{aligned} \tag{26}\]
Setting \[\xi := \eta_0 \,\mu_0^{-1} \in U_{r – \nu}\,,\] it is clear that in (26), the set of possible \(\eta_0, \mu_0\) consists of all units in \(U_{r-\nu}\) since \({\bf z}_1, {\bf z}_1'\) are independent elements of \({\mathbb Z}_q^n \setminus (p)^{(n)}\) . As a result, the set of all possible \(\xi\) also equals \(U_{r-\nu}\) and (26) says the following: \[\label {CS 1 bisbis4} {\bf m}_2 \in \langle {\bf z}_1 \rangle \cap \langle {\bf z}_1' \rangle \ \mbox{ and } \ {\bf m}_2 := p^{\nu} \,\eta_0 \,{\bf z}_1 = p^{\nu} \, \mu_0 \, {\bf z}_1' \implies {\bf z}_1' = \xi \, {\bf z}_1 \ mod \, p^{r-\nu}. \tag{27}\]
In other words \[\begin{aligned} \label {CS 1 bisbisbis4} &{\bf m}_2 \in \langle {\bf z}_1 \rangle \cap \langle {\bf z}_1' \rangle \nonumber \\ &\implies \ \chi \langle \mathbf \zeta ({\bf z}_1), – {\bf m}_2 \rangle \cdot \chi \langle \mathbf \zeta' ({\bf z}_1'), {\bf m}_2 \rangle = \chi \left(p^{\nu} \,\mu_0 \, j \, (1 – \xi)\right) \ 1_{\{{\bf z}_1' = \xi \,{\bf z}_1 (p^{r-\nu})\}} \ ({\bf z}_1, {\bf z}_1'). \end{aligned} \tag{28}\]
Setting, for each \(\nu\) and \(\xi \in U_{r-\nu}\), \[N ( \nu, \xi) := \sum_{{\bf z}_1, {\bf z}_1' \notin (p)^{(n)}} \ 1_{\{{\bf z}_1' = \xi \,{\bf z}_1 (p^{r-\nu})\}} \ ({\bf z}_1, {\bf z}_1')\,,\] it is elementary to see that \[\label{value of N} N ( \nu, \xi) = c_{p, n} \, q^n \,p^{\nu n} \quad \mbox{ uniformly in $\xi \in U_{r-\nu}$}. \tag{29}\]
We now evaluate the contribution from those \({\bf z}_1 \neq {\bf z}_1'\) in (22) as follows: \[\begin{aligned} \label{unequal contribution} & \sum_{{\bf m}_2 \neq 0} \ \sum_{{\bf z}_1 \neq {\bf z}_1' \notin (p)^{(n)}} \chi \langle \mathbf \zeta ({\bf z}_1), -{\bf m}_2 \rangle \cdot \chi \langle \mathbf \zeta'({\bf z}_1'), {\bf m}_2 \rangle \cdot 1_{\langle {\bf z}_1 \rangle \cap \langle {\bf z}_1' \rangle} ({\bf m}_2) \nonumber \\ &\qquad = \sum_{0\le \nu \le r-1} \ \sum_{\mu, \xi \in U_{r – \nu}} \ \chi \left(p^{\nu} \, j \,\mu \, (1 – \xi) \right) \ N(\nu, \xi) \nonumber \\ &\qquad = \sum_{0\le \nu \le r-1} \ \sum_{\mu \in U_{r – \nu}} \ \chi \left(p^{\nu} \, j \,\mu\right) \, E_\nu (\mu) \ \end{aligned} \tag{30}\] where \[\begin{aligned} \label{unequal contribution bis} E_\nu (\mu) =& \sum_{\xi \in U_{r-\nu}} \chi (- p^{\nu} \, j\, \mu \, \xi) \, N (\nu, \xi) \nonumber \\ =& c_{p, n} \, q^n \, p^{\nu n}\, \sum_{\xi \in U_{r-\nu}} \chi (- p^{\nu} \, j\, \mu \, \xi) \nonumber \\ =& c_{p, n} \, q^n \, p^{\nu n}\, \sum_{\kappa \in U_{r-\nu}} \chi (p^{\nu} \, j\, \kappa)\,, \end{aligned} \tag{31}\] where the last line is due to the fact that for each fixed \(\mu,\) \(\{\mu \, \xi : \xi \in U_{r-\nu}\} = U_{r – \nu}\).
It is a standard fact (see [3], pg. 56) that there is considerable cancellation in the character sums as follows: \[\label {character sum} \sum_{\kappa \in U_{r-\nu}} \chi (p^{\nu} \, j\, \kappa) = \begin{cases} -1 \ &\mbox{ if } \nu = r-1, \\ 0 \ &\mbox{ if } \nu \le r -2. \end{cases} \tag{32}\]
Combining (30), (31) with (32) amounts to evaluating (30) at \(r = n -1\). Then, combining this with (25), now tells us the following: \[\begin{aligned} \label{evaluation} rhs (15) =& q^{-2 n – 2} \cdot \big\{C_{p, r, n} \, q^n \, p^{-(n – 1)} + c_{p, n} \,q^{2 n} \, p^{- n} \big\} \nonumber \\ =& q^{-2} \, p^{-n} \cdot O(1) \quad \mbox{ uniformly in $r \ge 2$ and $p \gg 1.$} \end{aligned} \tag{33}\]
As a result we have now shown \[\label{last point} \sum_{{\bf m}_2 \neq 0} |\widehat 1_j ( 0, {\bf m}_2)|^2 = q^{-2} \, p^{-n} \, O(1)\,. \tag{34}\]
Going back to the original error bound in (9) we conclude that there exists constants \(P_I, C_I\) such that for all \(r \ge 2\) \[ p \ge P_I \implies \mathcal E_I^* \le C_I \, q^n \, |E| \, \delta_E^{\frac 12} \, q^{-1} \, p^{-\frac n2} = C_I \, q^{\frac n2 – 1} p^{-\frac n2} \, |E|^{\frac 32}. \ \tag{35}\]
In this way we see some improvement in the upper bound from (13), having eliminated the \(p^{1/4}\) factor.
The method is similar to that in Section 3.2.2, but the estimate we get (see (50)) is rather different in form to the bound in (35). The reason for this seems to be due to the fact that the behavior of the factor in the summation over \({\bf x}_2\) in (44) depends upon \(ord_p\, (1 -\xi)\), i.e., it is evidently not uniform in the variable \(\xi\).
As a result, it will suffice to emphasize in this section the details that differ from those in the preceding section (see Remark 2.2 in particular).
Defining (see (4)) for each \({\bf m}_1 \neq 0\) \[\label{F def} \mathcal F ({\bf m}_1) = \sum_{{\bf m}_2 \in {\mathbb Z}_q^n} \widehat 1_E ({\bf m}_2) \, \widehat 1_j ({\bf m}_1, {\bf m}_2), \tag{36}\] an application of Cauchy-Schwarz and Plancherel’s Theorem, as with \(\mathcal E_I^*\), first tells us that \[\begin{aligned} \label{E2 A} \mathcal E_{II}^* \le& q^{2 n} \cdot \left(\sum_{{\bf m}_1 \neq 0} |\widehat 1_E ({\bf m}_1)|^2 \right)^{1/2} \cdot \left(\sum_{{\bf m}_1 \neq 0} |\mathcal F ({\bf m}_1)|^2 \right)^{1/2} \nonumber \\ \le& q^{2 n} \cdot \delta_E^{1/2} \cdot \left(\sum_{{\bf m}_1 \neq 0} \ \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} \right)^{1/2}\,. \end{aligned} \tag{37}\]
The second step is to rewrite, for each \({\bf m}_1 \neq 0\), the factors in the sum (37) using the Fourier transform definition. This gives the following, in which we have set \(\pmb z := ({\bf z}_1, {\bf z}_2), \pmb z' := ({\bf z}_1', {\bf z}_2')\), and used the fact that \(\overline {\mathcal F ({\bf m}_1)} = \sum_{{\bf m}_2'} \widehat 1_E (-{\bf m}_2') \, \widehat 1_j (-{\bf m}_1, -{\bf m}_2')\): \[\begin{aligned} \label {E2 B} F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} =& q^{- 6 n} \cdot \sum_{{\bf m}_2, {\bf m}_2'} \bigg\{\left(\sum_{{\bf x}_2} 1_E ({\bf x}_2) \, \chi \langle {\bf x}_2, – {\bf m}_2 \rangle \right) \cdot \left(\sum_{{\bf x}_2'} 1_E ({\bf x}_2') \, \chi \langle {\bf x}_2', {\bf m}_2' \rangle \right) \nonumber \\ & \cdot \left(\sum_{\pmb z} 1_j (\pmb z) \, \chi \langle \pmb z, -({\bf m}_1, {\bf m}_2) \rangle \right) \cdot \left(\sum_{\pmb z'} 1_j (\pmb z') \, \chi \langle \pmb z', ({\bf m}_1, {\bf m}_2') \rangle \right)\bigg\}. \end{aligned} \tag{38}\]
In the third step we interchange the sums over \({\bf m}_2, {\bf m}_2'\) with those over \({\bf x}_2, \pmb z, {\bf x}_2', \pmb z'.\) This then gives: \[\begin{aligned} \label {E2 C} \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} =& q^{- 6 n} \cdot \bigg\{\sum_{{\bf x}_2, \pmb z} 1_E ({\bf x}_2) \, 1_j (\pmb z) \, \chi \langle {\bf z}_1, – {\bf m}_1 \rangle \cdot \left(\sum_{{\bf m}_2} \, \chi \langle {\bf x}_2 + {\bf z}_2, -{\bf m}_2 \rangle \right) \nonumber \\ &\cdot \sum_{{\bf x}_2', \pmb z'} 1_E ({\bf x}_2') \, 1_j (\pmb z') \, \chi \langle {\bf z}_1', {\bf m}_1 \rangle \cdot \left(\sum_{{\bf m}_2'} \chi \langle {\bf x}_2' + {\bf z}_2', {\bf m}_2' \rangle \right)\,. \end{aligned} \tag{39}\]
Since the sums over \({\bf m}_2, {\bf m}_2'\) are complete exponential sums over \({\mathbb Z}_q^n\), it now follows that \[\sum_{{\bf m}_2} \chi \langle {\bf x}_2 + {\bf z}_2, -{\bf m}_2 \rangle = q^n \cdot 1_{\{{\bf x}_2 + {\bf z}_2 = 0\}} ({\bf x}_2, {\bf z}_2),\] and \[\sum_{{\bf m}_2'} \chi \langle {\bf x}_2' + {\bf z}_2', {\bf m}_2' \rangle = q^n \cdot 1_{\{{\bf x}_2' + {\bf z}_2' = 0\}} ({\bf x}_2', {\bf z}_2')\,.\]
Substituting the right sides into (39) and simplifying a bit then shows: \[\begin{aligned} \label {E2 D} \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} = q^{- 4 n} \cdot II ({\bf m}_1) \cdot II' ({\bf m}_1), \end{aligned} \tag{40}\] where \[II ({\bf m}_1) := \sum_{{\bf x}_2, {\bf z}_1} 1_E ({\bf x}_2) \, 1_j ({\bf z}_1, – {\bf x}_2) \, \chi \langle {\bf z}_1, – {\bf m}_1 \rangle ,\] and \[II' ({\bf m}_1) := \sum_{{\bf x}_2', {\bf z}_1'} 1_E ({\bf x}_2') \, 1_j ({\bf z}_1', -{\bf x}_2') \, \chi \langle {\bf z}_1', {\bf m}_1 \rangle\,. \]
We can now compare these expressions with those in (14) since the right sides can be rewritten as follows \[\begin{aligned} \sum_{{\bf x}_2, {\bf z}_1} 1_E ({\bf x}_2) \, 1_j ({\bf z}_1, – {\bf x}_2) \, \chi \langle {\bf z}_1, – {\bf m}_1 \rangle =& \sum_{{\bf x}_2} 1_E ({\bf x}_2) \cdot \left(\sum_{{\bf z}_1} 1_j ({\bf z}_1, – {\bf x}_2) \, \chi \langle {\bf z}_1, – {\bf m}_1 \rangle \right) \\ \sum_{{\bf x}_2', {\bf z}_1'} 1_E ({\bf x}_2') \, 1_j ({\bf z}_1', – {\bf x}_2') \, \chi \langle {\bf z}_1', {\bf m}_1 \rangle =& \sum_{{\bf x}_2'} 1_E ({\bf x}_2') \cdot \left(\sum_{{\bf z}_1'} 1_j ({\bf z}_1', -{\bf x}_2') \, \chi \langle {\bf z}_1', {\bf m}_1 \rangle \right)\,. \end{aligned}\]
Adapting the idea from Section 2.2.2, for each \({\bf x}_2, {\bf x}_2',\) we note that the sums over \({\bf z}_1, {\bf z}_1'\) are, by definition, concentrated on the following two affine subspaces of \({\mathbb Z}_q^n\), which can only be nonempty when \({\bf x}_2, {\bf x}_2' \notin (p)^{(n)}\) since \(j \in U_q\) : \[\begin{aligned} \mathcal H_j ({\bf x}_2) :=& \{{\bf z}_1 : \langle {\bf z}_1, -{\bf x}_2 \rangle = j\} = \widetilde {\bf z}_1 ({\bf x}_2) + \mathcal K ({\bf x}_2);\\ \mathcal H_j' ({\bf x}_2') :=& \{{\bf z}_1' : \langle {\bf z}_1', – {\bf x}_2' \rangle = j\} = \widetilde{\bf z}_1' ({\bf x}_2') + \mathcal K ({\bf x}_2')\,, \end{aligned}\] where \(\mathcal K({\bf x}_2)\) resp. \(\mathcal K ({\bf x}_2')\) denotes the linear subspaces on which \({\bf z}_1 \to \langle {\bf z}_1, -{\bf x}_2 \rangle\) resp. \({\bf z}_1' \to \langle {\bf z}_1', -{\bf x}_2' \rangle\) vanishes, and \(\widetilde {\bf z}_1 ({\bf x}_2)\) resp. \(\widetilde {\bf z}_1' ({\bf x}_2')\) denotes a particular solution of \[\langle {\bf z}_1, -{\bf x}_2 \rangle = j \ \ \mbox{ resp.} \ \ \langle {\bf z}_1', – {\bf x}_2' \rangle = j\,.\]
Exactly as in (19), (21) \[\label{expsum vbis} \pmb x \in \{{\bf x}_2, {\bf x}_2'\} \implies \sum_{{\bf w}\in \mathcal K (\pmb x)} \ \chi \langle {\bf w}, {\bf v}\rangle = \begin{cases} q^{n-1} &\mbox{ if ${\bf v}\in \langle \pmb x \rangle $} \\ 0 &\mbox{ \ if not}. \end{cases} \tag{41}\]
When \({\bf v}= \pm {\bf m}_1\) we apply this for any \({\bf x}_2, {\bf x}_2'\) and conclude: \[\begin{aligned} \label{E2 E} \sum_{{\bf z}_1} 1_j ({\bf z}_1, – {\bf x}_2) \, \chi \langle {\bf z}_1, – {\bf m}_1 \rangle =& q^{n-1} \cdot \chi \langle \widetilde {\bf z}_1 ({\bf x}_2), – {\bf m}_1 \rangle \cdot 1_{ \langle {\bf x}_2 \rangle} ({\bf m}_1)\,,\nonumber\\ \sum_{{\bf z}_1'} 1_j ({\bf z}_1', -{\bf x}_2') \, \chi \langle {\bf z}_1', {\bf m}_1 \rangle =& q^{n-1} \cdot \chi \langle \widetilde {\bf z}_1' ({\bf x}_2'), {\bf m}_1 \rangle \cdot 1_{ \langle {\bf x}_2' \rangle} ({\bf m}_1)\,. \end{aligned} \tag{42}\]
Given that \(\nu := ord_p \, {\bf m}_1 \in [0, r – 1]\) we follow the same method from Section 2.2.2 by using the fact that \(ord_p \, {\bf x}_2 = ord_p \, {\bf x}_2' = 0\). Thus, there exist units \(\eta = \eta ({\bf m}_1, {\bf x}_2), \ \mu = \mu ({\bf m}_1, {\bf x}_2') \in U_{r – \nu}\) and \(\widehat {\bf m}_1 \in {\mathbb Z}_{r-\nu}^n\) with \(ord_p \,\widehat {\bf m}_1 = 0\) such that \[\begin{aligned} &\text{(i)} \ \ {\bf m}_1 \in \langle {\bf x}_2 \rangle \cap \langle {\bf x}_2' \rangle \iff \nonumber \\ &\quad \ \ \ \widehat {\bf m}_1 = \eta \, {\bf x}_2 \ mod \, p^{r-\nu};\ \ \widehat {\bf m}_1 = \mu \, {\bf x}_2' \ mod \, p^{r-\nu}; \ {\bf x}_2' = \mu \, \eta^{-1} {\bf x}_2 \ mod \,p^{r-\nu}; \nonumber \\ &\text{(ii)} \ \ \chi \langle \widetilde {\bf z}_1 ({\bf x}_2) , – {\bf m}_1 \rangle = \chi \left(- p^{\nu} \,\eta \, j \right) \quad \mbox{ and } \quad \chi \langle \widetilde {\bf z}_1' ({\bf x}_2') , {\bf m}_1 \rangle = \chi \left(p^{\nu} \, \mu \, j \right)\,. \nonumber \end{aligned}\]
Defining next \[\xi = \eta\, \mu^{-1} \in U_{r – \nu}\,,\] it will also be convenient (see Remark 2.3 below) to think of \(\xi\) as a unit in \(U_q\) as well by setting \[\label {canonical lift} \widehat \xi = \xi_0 + p \xi_1 + \cdots + p^{r-\nu-1} \xi_{r-\nu-1} + (p^{r-\nu} 0 + \cdots + p^{r-1} 0). \tag{43}\]
Such a unit of \(U_q\) can be thought of as the “canonical lift” of \(\xi\) to \(U_q.\)
We then combine (40) with (i), (ii) and the other preliminary remarks, to understand more precisely the sum over \({\bf m}_1 \neq 0\) in (37). Arguing as in Section 2.2.2, we have the following: \[\begin{aligned} \label {E2 F} \sum_{{\bf m}_1 \neq 0} \ \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} &= q^{- 4 n} \cdot \sum_{{\bf m}_1 \neq 0} \ \left\{\sum_{{\bf x}_2, {\bf x}_2'} 1_E ({\bf x}_2) \, 1_E ({\bf x}_2') \left(\sum_{{\bf z}_1} 1_j ({\bf z}_1, -{\bf x}_2) \chi \langle {\bf z}_1, -{\bf m}_1 \rangle \right)\right.\\ &\qquad\left. \cdot \left(\sum_{{\bf z}_1'} 1_j ({\bf z}_1', -{\bf x}_2') \chi \langle {\bf z}_1', {\bf m}_1 \rangle \right) \right\} \nonumber \\ &= q^{- 2 n – 2} \cdot \sum_{\nu = 0}^{r-1} \ \sum_{\eta, \mu \in U_{r-\nu}} \ \left\{\sum_{\substack {{\bf x}_2, {\bf x}_2' \notin (p)^{(n)} \\ {\bf x}_2' = \mu \eta^{-1}\, {\bf x}_2 \,(p^{r-\nu})}} 1_E ({\bf x}_2) \, 1_E ({\bf x}_2') \chi \left(p^{\nu } \, j \,(\mu – \eta) \right) \right\} \nonumber \\ &= q^{- 2 n – 2} \cdot \sum_{\nu = 0}^{r-1} \ \sum_{\mu, \xi \in U_{r-\nu}} \ \left\{\sum_{\substack {{\bf x}_2, {\bf x}_2' \notin (p)^{(n)} \\ {\bf x}_2' = \xi {\bf x}_2 \,(p^{r-\nu})}} 1_E ({\bf x}_2) \, 1_E ({\bf x}_2') \, \chi \left(p^{\nu } \,\mu \, j \, (1 – \xi) \right) \right\} \nonumber \\ &= q^{- 2 n – 2} \cdot \sum_{\nu = 0}^{r-1} \ \sum_{ \xi \in U_{r-\nu}} \ N_E(\nu, \xi) \,\mathcal E (\nu, \xi)\,. \end{aligned} \tag{44,45}\] where \[N_E(\nu, \xi) = \sum_{\substack {{\bf x}_2, {\bf x}_2' \notin (p)^{(n)} \\ {\bf x}_2' = \xi {\bf x}_2 \,(p^{r-\nu})}} 1_E ({\bf x}_2) \, 1_E ({\bf x}_2') \ \ \ \mbox{and} \ \ \ \mathcal E (\nu, \xi) := \sum_{ \mu \in U_{r-\nu}} \ \chi \left(p^{\nu} \, j \,\mu \, (1 – \xi) \right)\,.\]
Unlike the argument in Section 2.2.2, there is no uniform in \(\xi\) bound for the double sum over \({\bf x}_2, {\bf x}_2'\).
To overcome this issue, we must first sum over \(\mu\) for each fixed \(\xi.\) To do this, we first set, for any \(\xi \in U_{r-\nu},\) \[\rho (\xi) = 1 – \xi \in {\mathbb Z}_{p^{r-\nu}}\,; \ \ \kappa = ord_p \, \rho (\xi)\,; \ \ \rho (\xi) = p^\kappa \,\widehat \rho(\xi) \ \ \ \mbox{ where } \ \ \widehat \rho(\xi) \in U_{r – \nu – \kappa}\,,\] where it is understood that \(\kappa \le r – \nu\) with equality iff \(\xi = 1.\) We then break up the sum over \(\xi\) according to the value of \(\kappa\). It then follows that \[\begin{aligned} \label{E2 F bisbis} &\sum_{{\bf m}_1 \neq 0} \ \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} = q^{- 2 n – 2} \sum_{0 \le \nu \le r-1} \ \sum_{\kappa \le r-\nu} \ \sum_{\substack{\xi \in U_{r-\nu} \\ ord_p \, \rho (\xi) = \kappa}} N_E(\nu, \xi) \, \mathcal E (\nu, \kappa, \xi) \end{aligned} \tag{46}\] where\[\mathcal E (\nu, \kappa, \xi) = \sum_{\mu \in U_{r – \nu}} \ \chi \left(p^{\nu + \kappa} \, j \, \mu \, \widehat \rho (\xi) \right)\,.\]
It is now evident that the character sum evaluations in (32) apply to each \(\mathcal E (\nu, \kappa, \xi)\). In particular, for given \(\nu\) there is a nonzero contribution only when \(\kappa \in \{r – \nu – 1, r – \nu\}\), in which event it follows that \[(44) = q^{-2 n – 2} \sum_{0 \le \nu \le r-1} \left(A(\nu) + B(\nu)\right)\] where \[\begin{aligned} A(\nu) = & \sum_{\substack {\xi \in U_{r-\nu} \\ \xi = 1 + p^{r-\nu-1} \hat \rho_{r-\nu-1} (\xi) \\ \hat \rho_{r-\nu-1} (\xi) \in U_{1}}} \ N_E (\nu, \xi) \sum_{\mu \in U_{r – \nu}} \ \chi \left(p^{r – 1} \, j \, \mu \, \widehat \rho_{r-\nu-1} (\xi) \right) \nonumber \\ =& (- q \, p^{-\nu – 1})\, \sum_{\substack {\xi \in U_{r-\nu} \\ \xi = 1 + p^{r-\nu-1} \hat \rho_{r-\nu-1} (\xi) \\ \hat \rho_{r-\nu-1} (\xi) \in U_{1}}} \ N_E (\nu, \xi) \quad (\mbox{ $\sum_{\mu \in U_{r-\nu}} (\cdot)$ is uniform in $\xi$}),\\ B(\nu) =& |U_{r-\nu}| \, \sum_{\substack {{\bf x}_2, {\bf x}_2' \notin (p)^{(n)} \\ {\bf x}_2' = {\bf x}_2 \,(p^{r-\nu})}} 1_E ({\bf x}_2) \, 1_E ({\bf x}_2') \le c_p \,q \, p^{\nu (n – 1)}\, |E|\,. \nonumber \end{aligned}\]
For \(\nu \ge 1\) we denote the coset mod \(p^{r-\nu}\) of any vector \({\bf z}\in {\mathbb Z}_q^n\) by \[[{\bf z}]_{r-\nu} = \{{\bf y}\in {\mathbb Z}_q^n : {\bf y}= {\bf z}\, (p^{r-\nu})\}\,.\]
Then, for each \({\bf x}_2 \notin (p)^{(n)}\) and \(\hat \xi \in U_q\), the canonical lift of \(\xi \in U_{r-\nu}\) (see (43)), it follows that \[\begin{aligned} \label {A bound} \sum_{\substack {\xi \in U_{r-\nu} \\ \xi = 1 + p^{r-\nu-1} \hat \rho_{r-\nu-1} (\xi) \\ \hat \rho_{r-\nu-1} (\xi) \in U_{1}}} \ N_E (\nu, \xi) :=& \sum_{{\bf x}_2 \notin (p)^{(n)}} 1_E ({\bf x}_2) \left(\sum_{\substack {\xi \in U_{r-\nu} \\ \xi = 1 + p^{r-\nu-1} \hat \rho_{r-\nu-1} (\xi) \\ \hat \rho_{r-\nu-1} (\xi) \in U_{1}}} \ \sum_{{\bf z}\in [\hat \xi \,{\bf x}_2]_{r-\nu}}1_E ({\bf z}) \right) \nonumber\\ \le& c_p \, p^{n \nu + 1} \, |E| \quad (\textit{since } |U_1| = p – 1). \end{aligned} \tag{47}\]
This extends trivially to the case \(\nu = 0\) since in that event each congruence \({\bf x}_2' = \xi \, {\bf x}_2\, (p^r)\) has but one solution. From this it is easy to verify that \[\begin{aligned} \label{E2 G} \sum_{0 \le \nu \le r-1} B (\nu) \le& c_p \,q \, |E| \, \sum_{\nu = 0}^{r-1} \,p^{\nu (n – 1)} = q^n \, p^{-(n-1)} \, |E| \, O(1) \nonumber \\ \left|\sum_{0\le \nu \le r-1} A(\nu)\right| \le \sum_{0 \le \nu \le r-1} |A(\nu)| \le& q \, p^{-1} \, |E| \, \sum_{\nu = 0}^{r-1} \,p^{\nu (n – 1)} = q^n \, p^{-n} \, |E| \, O(1)\,. \end{aligned} \tag{48}\]
As a result, we have shown the existence of a constant \(C_{II}\) which is uniform in \(p \gg 1\) and \(r \ge 2\) such that \[\label {E2 L} \left(\sum_{{\bf m}_1 \neq 0} \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} \right)^{1/2} \le C_{II} \,q^{-\frac n2 – 1}\,p^{-\frac {n – 1}2} \, |E|^{\frac 12}\,, \tag{49}\] from which (37) now implies:
There exist constants \(P_{II}\) and \(C_{II}\) such that for all \(r \ge 2\) and \(j\): \[\label{E2 M} p \ge P_{II} \implies \mathcal E_{II}^* \le C_{II} \, q^{n-1} \, p^{- \frac {n-1}2} \, |E|\,. \tag{50}\]
Remark 2.2. The fact that this argument gives the factor \(|E|\) in (50), not \(|E|^{3/2}\) (as in (13)), is crucial for the proof of Theorem 1.1
Remark 2.3. One could only hope to improve upon this if one knew something more precise about the function \[\xi \longrightarrow |\{{\bf x}' \in E : {\bf x}' = \xi {\bf x}\,(p^{r-\nu}) \ \mbox{for some ${\bf x}\in E$} \}|\,.\]
To that end, and with a sharpening in mind (see Section 3) whenever \(E\) is replaced by \[E_v := E \cap S_{v} \qquad \mbox{ for some $v \in U_q$}\,,\] which will be used in Section 3, it is useful to point out here how (50) can be improved in that event.
We first note that (see (47)) for any \(1 \le \nu \le r – 1\) and \(\xi \in U_{r-\nu}\) such that \(1 – \xi = p^{r – \nu – 1} \, \widehat \rho (\xi)\) with \(\widehat \rho (\xi) \in U_{1}\) : \[\begin{aligned} \label{xi mod q} &{\bf x}_2' = \xi \, {\bf x}_2 \, (p^{r-\nu}) \ \mbox{ and } \ 1_{E_v} ({\bf x}_2') = 1_{E_v} ({\bf x}_2) = 1 \\ &\implies {\bf x}_2' \in [\hat \xi \,{\bf x}_2]_{r-\nu} = \{\hat \xi \, {\bf x}_2 + p^{r – \nu} {\bf y}: {\bf y}= {\bf y}_0 + p\, {\bf y}_1 + \cdots + p^{\nu-1} {\bf y}_{\nu – 1} \in {\mathbb Z}_{p^\nu}^n\} \nonumber \nonumber \\ &\implies v (1 – \hat \xi^2) = p^{r – \nu} \left(2\, \hat \xi \, \langle {\bf x}_2, {\bf y}\rangle + p^{r – \nu} \|{\bf y}\| \right) \qquad (\mbox{ an equation in $ {\mathbb Z}_q$})\,.\nonumber \end{aligned} \tag{51}\] As a result, if \({\bf x}_2' = \hat \xi \,{\bf x}_2 + p^{r – \nu} {\bf y}\) satisfies \(1_{E_v} ({\bf x}_2') = 1\) then \(p^{r – \nu} | v (1 – \hat \xi^2)\) in \({\mathbb Z}_q\).
On the other hand, since \(\xi \neq 1 \, (p^{r-\nu})\) it follows that \[\hat \xi^2 = (1 – p^{r-\nu-1} \hat \rho (\xi))^2 = 1 – 2 \hat \rho (\xi)\, p^{r-\nu-1} \ (p^{r – \nu}) \neq 1 \ (p^{r-\nu}).\]
In other words, it is not possible that \[v (1 – \hat \xi^2) = 0 \, (p^{r-\nu}).\]
This now tells us that whenever \(E\) is replaced by \(E_v\) in (36) then \[(44) = q^{-2 n – 2} \sum_{0 \le \nu \le r-1} \left(A(\nu) + B(\nu)\right) = \sum_{0 \le \nu \le r – 1} B(\nu) \,,\] that is, only each \(B(\nu)\), when \(\xi = 1 \, (p^{r-\nu})\) can effectively contribute.
From this we now conclude that any \({\bf y}\) appearing in (51) must then satisfy the congruence \[\label {algebraic improvement} 0 = 2 \, \langle {\bf x}_2, {\bf y}\rangle + p^{r – \nu} \|{\bf y}\| \ (p^{\nu}). \tag{52}\]
This condition determines a non singular hypersurface in \({\mathbb Z}_{p^\nu}^n\) since the linear part is not identically zero, given that \({\bf x}_2 \notin (p)^{(n)}\).
Applying the standard method, via Hensel’s Lemma, of lifting solutions mod \(p\) to solutions mod \(p^{\nu}\), which is possible because the linear part \({\bf y}\to \langle {\bf x}_2, {\bf y}\rangle\) is non zero mod \(p\), it follows that \[\label {algebraic improvement bis} \big|\{{\bf y}\in {\mathbb Z}_{p^\nu}^n : (52) \mbox{ holds }\} \big| = O(p^{\nu (n – 1)}). \tag{53}\]
As a result, for any \(v \in U_q\), when we replace \(E\) by \(E_v\) in the definition (36) of \(\mathcal F ({\bf m}_1)\), (47) – (49) can then be improved as follows: \[\begin{aligned} \label {A boundbis} \sum_{0 \le \nu \le r-1} B (\nu) \le c_p \,q \, | E_v\,| \, \sum_{\nu = 0}^{r-1} \,p^{\nu (n – 2)} &= q^{n-1} \, p^{-(n-2)} \, | E_v | \, O(1) \nonumber \\ \left(\sum_{{\bf m}_1 \neq 0} \mathcal F({\bf m}_1) \cdot \overline {\mathcal F ({\bf m}_1)} \right)^{1/2} &\le C_{II} \,q^{-\frac n2 – \frac 32 }\,p^{-\frac {n – 2}2} \, | E_v |^{\frac 12} \ \mbox{ (when\ $E = E_v$ in (36)}\,. \end{aligned} \tag{54}\]
The proof of Theorem 1.1 is now very easy.
We must compare the bound for \(\mathcal E_I^*\) in (35) with that for \(\mathcal E_{II}^*\) from (50) when \(p\) is sufficiently large and \(r \ge 2\). To that end, we first observe : \[\label {case 1} C_{II} \, q^{n-1} \, p^{- \frac {n-1}2} \, |E| < C_I \, q^{\frac n2 – 1} p^{-\frac n2} \, |E|^{\frac 32} \ \ \ \mbox{ iff } \ \ \ \delta_E \ge p \, \left( \frac { C_{II}}{C_{I}} \right)^2\,. \tag{55}\]
Since \(\delta_E \le 1\) this cannot occur for \(p\) sufficiently large. So it suffices to understand under what conditions is it the case that \[\mathcal M^* > 2 \, C_{II} \, q^{n-1} \, p^{- \frac {n-1}2} \, |E| \ge 2 \,\mathcal E_{II}^* \ge \mathcal E_I^* + \mathcal E_{II}^* = \mathcal E^*.\]
It is clear that this occurs if \[\delta_E > 2 \, C_{II} \, p^{- \frac {n-1}2}.\]
This now completes the proof of the assertion: \[\label{case 1 conclusion} \mbox{ $\exists \ C > 0$ \, s.t. \ $p \gg 1$ \ and \ $\delta_E \ge C \, p^{-\frac {n-1}2} \implies \mathcal M^* > \mathcal E^*$ \ \ if \ $n, r \ge 2$.} \tag{56}\]
Recalling the definition of \(\mathbf \Sigma_{{\bf v}, j} (E)\) from the Introduction, it is clear that \[\begin{aligned} \label{sigma def} \sigma_{{\bf v}, j} (E) :=& |\,\mathbf \Sigma_{{\bf v}, j} (E)\,| = \sum_{\pmb x = ({\bf x}_1, {\bf x}_2) \in {\mathbb Z}_q^{2 n}} \ 1_{E, v_1} ({\bf x}_1) \cdot 1_{E, v_2} ({\bf x}_2) \cdot 1_j (\pmb x)\,, \end{aligned} \tag{57}\] where for each \(u\)\[1_{E, v_u} ({\bf x}_u) = 1_E ({\bf x}_u) \cdot 1_{S_{ v_u}} ( {\bf x}_u),\] is the characteristic function of \(E_{v_u} := E \cap S_{v_u}\).
Applying Fourier inversion as in Section 2.2.1, it follows that \[\begin{aligned} \label {sigma + fi} \sigma_{{\bf v}, j} (E)=& \sum_{\pmb m = ({\bf m}_1, {\bf m}_2)} \widehat 1_j (\pmb m) \ \sum_{\pmb x = ({\bf x}_1, {\bf x}_2) \in {\mathbb Z}_q^{2 n}} \ 1_{E, v_1} ({\bf x}_1) \cdot 1_{E, v_2} ({\bf x}_2) \cdot \chi \langle \pmb x, \pmb m \rangle \nonumber \\ =& q^{2 n} \cdot \sum_{\pmb m = ({\bf m}_1, {\bf m}_2)} \widehat 1_j (\pmb m) \cdot \widehat 1_{E, v_1} (-{\bf m}_1) \cdot \widehat 1_{E, v_2} (-{\bf m}_2) := \mathcal M_{{\bf v}, j}^* + \mathcal E_{{\bf v}, j}^* , \end{aligned} \tag{58}\] where (see (7)) \[\begin{aligned} \label{error terms bis} \mathcal M_{{\bf v}, j}^* :=& q^{2 n} \cdot \widehat 1_j ( 0, 0) \cdot \widehat 1_{E, v_1} ( 0) \cdot \widehat 1_{E, v_2} ( 0) = \frac {|E_{v_1}| \cdot |E_{v_2}|}q \cdot \left(1 + O(p^{-\frac{2 n – 1}2})\right)\nonumber \\ \mathcal E_{{\bf v}, j}^* :=& q^{2 n} \cdot \sum_{\pmb m \neq ( 0, 0)} \widehat 1_j (\pmb m) \cdot \widehat 1_{E, v_1} (-{\bf m}_1) \cdot \widehat 1_{E, v_2} (-{\bf m}_2) \nonumber \\ =& q^{2 n} \cdot \bigg\{\widehat 1_{E, v_1} ( 0) \cdot \sum_{{\bf m}_2 \neq 0} \widehat 1_j ( 0, {\bf m}_2) \cdot \widehat 1_{E, v_2} (-{\bf m}_2) \nonumber \\ & \phantom{………} + \sum_{{\bf m}_1 \neq 0} \sum_{{\bf m}_2} \widehat 1_j (\pmb m) \cdot \widehat 1_{E, v_1} (-{\bf m}_1) \cdot \widehat 1_{E, v_2} (-{\bf m}_2)\bigg\} \nonumber \\ :=& q^{2 n} \cdot \bigg[\delta_{E_{v_1}} \cdot \sum_{{\bf m}_2 \neq 0} I_{v_2, j} ({\bf m}_2) + \sum_{{\bf m}_1 \neq 0} \sum_{{\bf m}_2} II_{{\bf v}, j} (\pmb m)\bigg]\,. \nonumber \\ :=& \mathcal E^*_{I} + \mathcal E^*_{II}. \end{aligned} \tag{59}\]
The extension of (35) to bound \(\mathcal E_{I }^*\) is straightforward. It suffices to replace \(\widehat 1_E ({\bf m}_2)\) in the applications of Cauchy-Schwarz and Plancherel in (9) by \(\widehat 1_{E, v_2}\), in which event the factor \(\delta_E^{1/2}\) should change to \(\delta_{E_{v_2}}^{1/2}.\) Furthermore, the definitions of \(P_{I}, c_I\) in Remark 1 remain unchanged.
We then know that \[\label{EI* boundbis1} p \ge P_I \implies \mathcal E_{I }^* \le C_I \, q^n \, |E_{v_1}| \, \delta_{E_{v_2}}^{\frac 12} \, q^{-1} \, p^{-\frac n2} = C_I \, q^{\frac n2 – 1} p^{-\frac n2} \, |E_{v_1}||E_{v_2}|^{\frac 12} := \mathcal B_{I}^*\,. \tag{60}\]
Moreover, unlike the situation in Section 2.2.1, where the choice is made to fix \({\bf m}_1 = 0\) and leads to a bound that is independent of this choice, in this section there are actually two possible choices to be made of that index \(\iota\) for which \({\bf m}_\iota = 0\). In other words, there is a second possible bound as follows: \[\label{EI* boundbis2} p \ge P_I \implies \mathcal E_{I}^* \le C_I' \, q^n \, |E_{v_2}| \, \delta_{E_{v_1}}^{\frac 12} \, q^{-1} \, p^{-\frac n2} = C_I' \, q^{\frac n2 – 1} p^{-\frac n2} \, |E_{v_2}||E_{v_1}|^{\frac 12} := \widetilde {\mathcal B}_{I}^*\,. \tag{61}\]
The extension of (50) to bound \(\mathcal E_{II}^*\), given that one has decided to fix \({\bf m}_1 = 0\) to define \(\mathcal E_{I}^*\), is also straightforward and similarly achieved by replacing \(\widehat 1_E ({\bf m}_1)\) resp. \(\widehat 1_E ({\bf m}_2)\) wherever either function appears in the discussion of Section 2.2.3 by \(\widehat 1_{E, v_1} ({\bf m}_1)\) resp. \(\widehat 1_{E, v_2} ({\bf m}_2)\). Applying the bounds (54) from Remark 2.3, we conclude as follows:
There exist \(P_{II}\) and \(C_{II}\) uniform in \(r \ge 2\) and \({\bf v}, j\) such that : \[\label{E2 Mbis} p \ge P_{II} \implies \mathcal E_{II}^* \le C_{II} \, q^{n-\frac 32} \,p^{-\frac{n-2}2}\, |E_{v_1}|^{\frac 12} \, |E_{v_2}|^{\frac 12} := \mathcal B_{II}^*\,. \tag{62}\]
Note that this bound is actually independent of the choice to fix \({\bf m}_1 = 0\) in the definition of \(\mathcal E_{I}^*\). That is, exactly the same estimate holds if we had chosen to define \(\mathcal E_{I}^*\) by setting \({\bf m}_2 = 0.\) The main point here is that the exponent of each \(|E_{v_u}|\) in \(\mathcal B_{II}^*\) equals \(1/2\) and is independent of the choice made to define \(\mathcal E_{I}^*\).
As a result, the single choice to make concerns whether \(\mathcal B_{I}^*\) is at least resp. at most \(\widetilde {\mathcal B}_{I}^*\). This is evidently equivalent to \[(a)\ \delta_{E_{v_1}} \ge \delta_{E_{v_2}} \quad \mbox{resp.} \quad (b) \ \delta_{E_{v_1}} \le \delta_{E_{v_2}}.\]
If (a) occurs, then \(\widetilde {\mathcal B}_{I}^*\) is better (i.e. smaller) than \(\mathcal B_{I}^*\), so we should next compare \(\widetilde {\mathcal B}_{I}^*\) to \(\mathcal B_{II}^*\). To that end we note that \[\mathcal B_{II}^* \le \widetilde {\mathcal B}_{I}^* \ \ \ \mbox{ iff } \ \ \ C_{II} \, q^{n-\frac 32} \,p^{-\frac{n-2}2}\, |E_{v_1}|^{\frac 12} \, |E_{v_2}|^{\frac 12} \le C_I' \, q^{\frac n2 – 1} p^{-\frac n2} \, |E_{v_2}||E_{v_1}|^{\frac 12}\,.\]
But since this inequality would require \[q^{-\frac 12} \, (1 + o(1)) = \delta_{S_{v_2}}^{\frac 12} \ge \delta_{E_{v_2}}^{\frac 12} \gg q^{-\frac 12} \, p,\] it cannot occur for \(p \gg 1.\)
As a result : \[\mbox{ for $p$ sufficiently large it is not possible that $\mathcal B_{II}^* \le \widetilde {\mathcal B}_{I}^*$.}\]
So, we may assume \(\mathcal B_{II}^* > \widetilde {\mathcal B}_{I}^*\) whenever \(p \gg 1\), in which event, it is now elementary to verify the following: \[\label{condition 1} p \gg 1\,, \ \ n, r \ge 2\,, \ \mbox{ and } \ \ \delta_{E_{v_1}}^{\frac 12} \ \delta_{E_{v_2}}^{\frac 12} > 2 \, C_{II} \,q^{-\frac 12}\, p^{-\frac {n-2}2} \ \ \ \mbox{ implies } \ \ \ \mathcal M_{{\bf v}, j}^* > 2 \, \mathcal B_{II}^* > \mathcal E_{{\bf v}, j}^*\,. \tag{63}\]
The lower bound in (63) however, must also be compatible with an a priori upper bound \[q^{-1}\, (1 + o(1)) \ge \delta_{S_{v_1}}^{\frac 12} \, \delta_{S_{v_2}}^{\frac 12} \ge \delta_{E_{v_1}}^{\frac 12} \, \delta_{E_{v_2}}^{\frac 12} > 2 \, C_{II} \,q^{-\frac 12}\, p^{-\frac {n-2}2}\,.\]
The two inequalities are evidently compatible if \[\label{constant size bound} 1+ o(1) > 2 C_{II} \, p^{\frac {r – (n-2)}2} \quad \mbox{ that is, if } \quad r < n – 2 \quad \mbox{ when } \ \ p \gg 1. \tag{64}\]
In other words, if \(\delta_{E_{v_1}} \ge \delta_{E_{v_2}}\), \(n \ge 2\), and \(r < n – 2\), we conclude that if \(p \gg 1\) then \[\label{final main term} q^{-2} (1 + o(1))^2 \ge \delta_{E_{v_1}} \, \delta_{E_{v_2}} > 4\, C_{II}^2 \, q^{-1} \, p^{2-n} \ \ \ \mbox{ implies} \ \ \ \sigma_{{\bf v}, j} (E) = \frac{|E_{v_1}| \, |E_{v_2}|}q \, (1 + o(1)). \tag{65}\]
If, however, \(\delta_{E_{v_1}} \le \delta_{E_{v_2}}\), then it is clear that it suffices to interchange \(v_1\) with \(v_2\) as subscripts in (63) and use the same argument as above to conclude that \(\mathcal M_{{\bf v}, j}^* > \mathcal E_{{\bf v}, j}^*\) whenever \(p \gg 1\) and \(r < n – 2.\) We leave these details for the reader to verify.
The union \(\Omega\) of the two sets of permissible densities is independent of \({\bf v}, j\) and satisfies the property that if \(E\) is any set for which \((\delta_{E_{v_1}}, \delta_{E_{v_2}}) \in \Omega\), then \(\mathcal M_{{\bf v}, j}^* > \mathcal E_{{\bf v}, j}^*\) and \[\sigma_{{\bf v}, j} (E) = \frac {|E_{v_1}| \, |E_{{\bf v}_2}|}q \cdot (1 + o(1)) \quad \mbox{ uniformly in $p \gg 1$ whenever $2 \le r < n – 2$ and $n \ge 5$}. \]
It is natural to want to apply our methods to detect the presence of particular dot product values between \(n-\)vectors over the ring of \(p-\)adic integers \(\mathcal Z_p\) when \(n \ge 2.\) We also continue to use the notation \((p)^{(n)}\) to denote the \(n-\)fold product of the maximal ideal \((p)\) in \(\mathcal Z_p.\)
Starting with a subset \(\mathcal E \subset \mathcal Z_p^n \setminus (p)^{(n)}\) and denoting by \(E_r\) the projection (or truncation by \(p^r\)) of \(\mathcal E\) to \({\mathbb Z}_q^n,\) we impose the hypotheses:
(i) \(p \gg 1 \ \mbox{ and } \ \ \delta_{E_r} \gg \ p^{- \frac {n – 1}2} \quad \mbox{ for all $r \ge 1$};\)
(ii) \(\mathcal E = \varprojlim_r E_r.\)
Note that condition (i) is a purely Haar measure theoretic condition since the density of \(E_r \subset {\mathbb Z}_q^n\) equals the normalized Haar measure of \(E_r.\) Condition (ii) says that \(\mathcal E\) equals the intersection of its family of approximating “tubular” neighborhoods \(\mathcal T_r (\mathcal E)\) of width \(p^{-r n}\): \[\mathcal T_r (\mathcal E) := \{ {\bf z}\in \mathcal Z_p^n: max_{{\bf x}\in \mathcal E} \ \|{\bf z}- {\bf x}\| \le p^{-r n}\}.\]
Remark 4.1. Note that this property does not hold if \(\mathcal E = {\mathbb Z}^n.\)
Theorem 1.1 tells us that if the two hypotheses are satisfied, for example, if \(\mathcal E\) is a closed subset (in the \(p-\)adic metric topology), and if, in addition, \(\delta_{E_r} \gg \ p^{- \frac {n – 1}2}\) for each \(r\), then for any unit \({\bf j}\in \mathcal Z_p\) \[\beta_{j_r } (E_r) \neq \emptyset \qquad (\mbox{ where $j_r \in {\mathbb Z}_q$ denotes the reduction mod $q$ of ${\bf j}$})\,.\]
Denoting by \(({\bf x}_r, {\bf y}_r)\) a point in \(E_r \times E_r\) for which \(\langle {\bf x}_r, {\bf y}_r \rangle = j_r\) the resulting sequence of points \(\{({\bf x}_r, {\bf y}_r)\}_r\) is a Cauchy sequence in \(\mathcal Z_p^n \times \mathcal Z_p^n\) . By the hypothesis (ii), the unique limit point \(({\bf x}, {\bf y})\) of this sequence belongs to \(\mathcal E \times \mathcal E\). Moreover, since \({\bf j}= lim_r \, j_r\) it follows that \[\langle {\bf x}, {\bf y}\rangle = {\bf j}\,.\]
In other words, for any unit \({\bf j}\) we have shown that if the hypotheses (i), (ii) are satisfied by a set \(\mathcal E \subset \mathcal Z_p^n \setminus (p)^{(n)}\) then \(\{({\bf x}, {\bf y}) \in \mathcal E \times \mathcal E : \langle {\bf x}, {\bf y}\rangle = {\bf j}\} \neq \emptyset\,.\)