Related papers: Formalizing Singer Sidon Constructions and Sidon S…

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product set, AA+1 must have a large part outside of any generalized…

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $\sigma(n+1) = k\sigma(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation…

The aim of this article is to construct a specific Poisson transform mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. The transforms we construct have values…

We discuss four off-shell N=4 D=1 supersymmetry transformations, their associated one-dimensional sigma-models and their mutual relations. They are given by I) the (4,4)_{lin} linear supermultiplet (supersymmetric extension of R^4), II) the…

We provide a new upper bound for sampling numbers $(g_n)_{n\in \mathbb{N}}$ associated to the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants…

Let $P_k:= \mathbb{F}_2[x_1,x_2,\ldots ,x_k]$ be the polynomial algebra in $k$ variables with the degree of each $x_i$ being $1,$ regarded as a module over the mod-$2$ Steenrod algebra $\mathcal{A},$ and let $GL_k$ be the general linear…

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an…

Let $p<q$ be odd primes, $\rho_1$ and $\rho_2$ be irreducible representations of $\text{SL}(2,\mathbb{Z}_p)$ and $\text{SL}(2,\mathbb{Z}_q)$ of dimensions $\frac{p+1}{2}$ and $\frac{q+1}{2}$, respectively. We show that if…

The Sombor index, a degree-based topological descriptor introduced by Gutman in 2021, lacks closed-form expressions for complex hierarchical trees with multi-level pendant structures and nonuniform degree distributions, despite extensive…

For any fixed dimension $d \geq 3$ we construct a Nikodym set in $F_q^d$ of cardinality $q^d - (\frac{d-2}{\log 2} +1+o(1)) q^{d-1} \log q$ in the limit $q \to \infty$, when $q$ is an odd prime power. This improves upon the naive random…

We investigate the descriptional complexity of operations on semilinear sets. Roughly speaking, a semilinear set is the finite union of linear sets, which are built by constant and period vectors. The interesting parameters of a semilinear…

The Lie-Poisson algebra so(N+1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the ND spherical, Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly present a…

The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re(s)=\alpha$. The approach employs concepts of two-dimensional principal component analysis built on a…

The Lax pair formalism is considered to discuss the integrability of the N=1 supersymmetric sinh-Gordon model with a defect. We derive associated defect matrix for the model and construct the generating functions of the modified conserved…

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost…

It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. For each positive rational number $s$ we construct pairs of finitely presented groups $H\subset G$…

We study generating series encoding linking numbers between geodesics in arithmetic hyperbolic $3$-folds. We show that the series converge to functions on genus $2$ Siegel space and that certain explicit modifications have the…

We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon…

We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say $I_{\Lambda}(N,h)$, of the von Mangoldt function $\Lambda(n)$ ($:= \log p$ for prime-powers $n=p^r$, 0…