This work bridges the abstract representation theory of $GSp\left(4\right)$ with recent computational techniques. We construct four examples of paramodular newforms whose associated automorphic representations have local representations at $p=2$ that are supercuspidal. We classify all relevant irreducible, admissible, supercuspidal representations of $GSp\left(4,{\mathbb{Q}}_2\right)$, and show that our examples occur at the lowest possible paramodular level, 16. The required theoretical and computational techniques include paramodular newform theory, Jacobi restriction, bootstrapping and Borcherds products.

A generalized Beatty sequence is a sequence $V$ defined by $V\left(n\right)=p\lfloor n\alpha \rfloor +qn+r$, for $n=1,2,\dots$, where $\alpha$ is a real number, and $p,q,r$ are integers. Such sequences occur, for instance, in homomorphic embeddings of Sturmian languages in the integers.

We consider the question of characterizing pairs of integer triples $\left(p,q,r\right),\left(s,t,u\right)$ such that the two sequences $V\left(n\right)=\left(p\lfloor n\alpha \rfloor +qn+r\right)$ and $W\left(n\right)=\left(s\lfloor n\alpha \rfloor +tn+u\right)$ are complementary (their image sets are disjoint and cover the positive integers). Most of our results are for the case that $\alpha$ is the golden mean, but we show how some of them generalize to arbitrary quadratic irrationals.

We also study triples of sequences $V_i=\left(p_i\lfloor n\alpha \rfloor +q_{i}n+r_i\right)$, $i=1,2,3$ that are complementary in the same sense.

We prove various counting formulas for CM-types of CM-fields and use them to construct infinite families of degenerate CM-types.

We obtain a polynomial-time algorithm that, given input $\left(A,b\right)$, where $A=\left(B|N\right)\in {\mathbb{Z}}^{m\times n}$, $m<n$, with nonsingular $B\in {\mathbb{Z}}^{m\times m}$ and $b\in {\mathbb{Z}}^m$, finds a nonnegative integer solution to the system $Ax=b$ or determines that no such solution exists, provided that $b$ is located sufficiently “deep” in the cone generated by the columns of $B$. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.

As a discrete counterpart to the classical theorem of Fritz John on the approximation of symmetric $n$-dimensional convex bodies $K$ by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions $P\left(A,b\right)\subset {\mathbb{Z}}^n$ in order to cover (many of) the lattice points inside a convex body by a simple geometric structure. Among others, they proved that there exists a generalized arithmetic progressions $P\left(A,b\right)$ such that $P\left(A,b\right)\subset K\cap {\mathbb{Z}}^n\subset P\left(A,O{\left(n\right)}^{3n∕2}b\right)$. Here we show that this bound can be lowered to $n^{O\left(lnn\right)}$ and study some general properties of so called unimodular generalized arithmetic progressions.

Let $n>k>2$ be integers. Define a bipartite graph between all $k$-element and all $2$-element subsets of an $n$-element set by drawing an edge if and only if the first one contains the second. The domination number of this graph is determined up to a factor of $1+o\left(1\right)$. The short proof relies on some extremal results concerning hypergraphs.

We derive a new explicit formula for Bernoulli numbers in terms of the Stirling numbers of the second kind and the Euler numbers. As a corollary of our result, we obtain an explicit formula for the even Euler numbers in terms of the Stirling numbers of the second kind.

We modify the statement and proof of Theorem 1 in “Intersection theorems for $\left\{0,\pm 1\right\}$-vectors and $s$-cross-intersecting families”. A version of the paper that incorporates the errata is uploaded on the arXiv: https://arxiv.org/abs/1603.00938. We thank Danila Cherkashin and Sergei Kiselev for pointing out the error.