We show that for \(1\) separated subsets of \(\mathbb{R}^{2}\), the natural Marstrand type slicing statements are false with the counting dimension that was used earlier by Moreira and Lima and variants of which were introduced earlier in different contexts. We construct a \(1\) separated subset \(E\) of the plane which has counting dimension \(1\), while for a positive Lebesgue measure parameter set of tubes of width \(1\), the intersection of the tube with the set \(E\) has counting dimension \(1\). This is in contrast to the behavior of such sets with the mass dimension, in regards to slicing, where the slicing theorems hold true.

A certain residue representation of the inverse binomial coefficients makes them amenable to Egorychev method for the reduction of sums by analytic methods, wherein the main idea is to identify parts of the summands as residues of analytic functions. We illustrate the use of such residue representation on some instances varying in complexity, including a generalization of an identity by Sung Sik U and Kyu Song Chae in [13].

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 \(E\).

In this paper, we derive some new combinatorial inequalities by applying well known real analytic results like Hölder’s inequality, Young’s inequality, and Minkowiski’s inequality to the recursively defined sequence \(f_n\) of functions \[\begin{align*} f_0(x) & = \chi_{(-1/2, 1/2)} (x), \nonumber \\ f_{n+1}(x) & = f_n(x+1/2)+ f_n(x-1/2), n \in \mathbb{N}\,\cup \,\{0\}. \end{align*}\] Towards this goal, we derive the closed form of the aforementioned sequence \((f_n)_{n\in \mathbb{N}\,\cup \,\{0\}}\) of functions and show that it is a sequence of simple functions that are linear combinations of characteristic functions of some unit intervals \(I_{n,i},\, i=0,1, …, n\), with values the binomial coefficients \(\binom{n}{i}\) on each unit interval \(I_{n,i}\). We show that \(f_n \in L^p(\mathbb{R})),\, 1\leq p \leq \infty\). Besides applying real analytic methods to formulate some combinatorial inequalities, we also illustrate the application of some combinatorial identities. For example, we use the Vandermonde convolution (or Vandermonde identity), in the study of some properties of the sequence of functions \((f_n)_{n\in\mathbb{ N}\cup \{0\}}\). We show how the \(L^2\) norm of \(f_n\) is related to the Catalan numbers.

We study Lambert series generating functions associated with arithmetic functions \(f\), defined by
$$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m.$$
These expansions naturally generate divisor sums through Dirichlet convolution with the constant-one function and provide a useful framework for enumerating ordinary generating functions of many multiplicative functions in number theory. This paper presents an overview of key properties of Lambert series, together with combinatorial generalizations and a compendium of formulas for important special cases. The emphasis is on formal and structural aspects of the sequences generated by these series rather than on analytic questions of convergence. In addition to serving as an introduction, the paper provides a consolidated reference for classical identities, recent connections with partition-generating functions, and other useful Lambert series expansions arising in applications.