This paper presents a new construction of the \( m \)-fold metaplectic cover of \( \mathrm{GL}_n \) over an algebraic number field \( k \), where \( k \) contains a primitive \( m \)-th root of unity. A 2-cocycle on \( \mathrm{GL}_n(\mathbb{A}) \) representing this extension is given, and the splitting of the cocycle on \( \mathrm{GL}_n(k) \) is found explicitly. The cocycle is smooth at almost all places of \( k \). As a consequence, a formula for the Kubota symbol on \( \mathrm{SL}_n \) is obtained. The construction of the paper requires neither class field theory nor algebraic \( K \)-theory but relies instead on naive techniques from the geometry of numbers introduced by W. Habicht and T. Kubota. The power reciprocity law for a number field is obtained as a corollary.

Let \( \pi = \pi_1 \pi_2 \cdots \pi_n \) be any permutation of length \( n \), we say a descent \( \pi_i \pi_{i+1} \) is a {lower}, {middle}, {upper} if there exists \( j > i+1 \) such that \( \pi_j < \pi_{i+1}, \pi_{i+1} < \pi_j < \pi_i, \pi_i < \pi_j \), respectively. Similarly, we say a rise \( \pi_i \pi_{i+1} \) is a {lower}, {middle}, {upper} if there exists \( j > i+1 \) such that \( \pi_j < \pi_i, \pi_i < \pi_j < \pi_{i+1}, \pi_{i+1} < \pi_j \), respectively. In this paper, we give an explicit formula for the generating function for the number of permutations of length \( n \) according to the number of upper, middle, lower rises, and upper, middle, lower descents. This allows us to recover several known results in the combinatorics of permutation patterns as well as many new results. For example, we give an explicit formula for the generating function for the number of permutations of length \( n \) having exactly \( m \) middle descents.

We prove that a sumset of a TE subset of N (these sets can be viewed as “aperiodic” sets) with a set of positive upper density intersects any polynomial sequence. For WM sets (subclass of TE sets) we prove that the intersection has lower Banach density one. In addition we obtain a generalization of the latter result to the case of several polynomials.

In this paper, we prove the Tiling implies Spectral part of Fuglede’s cojecture for the three interval case. Then we prove the converse Spectral implies Tiling in the case of three equal intervals and also in the case where the intervals have lengths 1/2, 1/4, 1/4. Next, we consider a set Ω ⊂ R, which is a union of n intervals. If Ω is a spectral set, we prove a structure theorem for the spectrum provided the spectrum is assumed to be contained in some lattice. The method of this proof has some implications on the Spectral implies Tiling part of Fuglede’s conjecture for three intervals. In the final step in the proof, we need a symbolic computation using Mathematica. Finally with one additional assumption we can conclude that the Spectral implies Tiling holds in this case.

We show that if \( A \) is a finite subset of an abelian group with additive energy at least \( c|A|^3 \), then there is a set \( \mathcal{L} \subset A \) with \( |\mathcal{L}| = O(c^{-1} \log |A|) \) such that \( |A \cap \mathrm{Span}(\mathcal{L})| = \Omega(c^{1/3} |A|) \).

We provide further explanation of the significance of an example in a recent paper of Wolf in the context of the problem of finding large subspaces in sumsets.

Lucy Slater used Bailey’s \( {}_6\psi_6 \) summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper, we apply the same techniques to Chu’s \( {}_{10}\psi_{10} \) generalization of Bailey’s formula to produce quite general Bailey pairs. Slater’s Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater’s work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the \( {}_6\psi_6 \) summation formula.

Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation formulae (including special cases of the \( {}_6\psi_6 \) summation formula and Jackson’s \( {}_6\phi_5 \) summation formula) to derive some of our infinite products. We use the new Bailey pairs, and/or the summation methods mentioned above, to give new proofs of some general series-product identities due to Ramanujan, Andrews, and others. We also derive a new general series-product identity, one which may be regarded as a partner to one of the Ramanujan identities. We also find new transformation formulae between basic hypergeometric series, new identities of Rogers-Ramanujan type, and new false theta series identities. Some of these latter are a kind of “hybrid” in that one side of the identity consists of a basic hypergeometric series, while the other side is formed from a theta product multiplied by a false theta series. This type of identity appears to be new.

In [Fr2, Skr], Frolov and Skriganov showed that low discrepancy point sets in the multidimensional unit cube \([0,1)^s\) can be obtained from admissible lattices in \( \mathbb{R}^s \). In this paper, we get a similar result for the case of \( (\mathbb{F}_q((x^{-1})))^s \). Then we combine this approach with Halton’s construction of low discrepancy sequences.