Related papers: Korselt Rational Bases and Sets

We prove that a map onto a nilpotent group $Q$ has finitely generated kernel if and only if the preimage of the positive cone is coarsely connected as a subset of the Cayley graph for every full archimedean partial order on $Q$. In case $Q$…

We study number theoretic properties of the map $x \mapsto x^{x} \mod{p}$, where $x \in \{1,2,\ldots,p-1\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map…

We study the relationship between Bell states, finite groups and complete sets of bases. We show how to obtain a set of N+1 bases in which Bell states are invariant. They generalize the X, Y and Z qubit bases and are associated to groups of…

Let $\A\subset\binom{[n]}{r}$ be a compressed, intersecting family and let $X\subset[n]$. Let $\A(X)={A\in\A:A\cap X\ne\emptyset}$ and $\S_{n,r}=\binom{[n]}{r}({1})$. Motivated by the Erd\H{o}s-Ko-Rado theorem, Borg asked for which…

For a natural number $N\geq 2$ and a real $\alpha$ such that $0 < \alpha \leq \sqrt{N}-1$, we define $I_\alpha:=[\alpha,\alpha+1]$ and $I_\alpha^-:=[\alpha,\alpha+1)$ and investigate the continued fraction map $T_\alpha:I_\alpha \to…

The $n$-queens problem is to determine $\mathcal{Q}(n)$, the number of ways to place $n$ mutually non-threatening queens on an $n \times n$ board. We show that there exists a constant $\alpha = 1.942 \pm 3 \times 10^{-3}$ such that…

We study $\alpha$-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base $\alpha$, where $\alpha$ is an algebraic conjugate of a Pisot…

Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is…

We prove that for any integers $\alpha, \beta > 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ is decidable (where $\alpha^{\mathbb{N}}$…

Let $G$ denote a compact monothetic group, and let $$\rho (x) = \alpha_k x^k + \ldots + \alpha_1 x + \alpha_0,$$ where $\alpha_0, \ldots , \alpha_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the…

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by…

Let $q=p^\alpha$ be a fixed prime power, $k\geq 2$ be an integer. We give a new upper bound for the size of $k$-wise $q$-modular $L$-avoiding $L$-intersecting set systems, where $L$ is any proper subset of $\{0, \ldots , q-1\}$. Our proof…

Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or…

For an alternate base $\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$, we show that if all rational numbers in the unit interval $[0,1)$ have periodic expansions with respect to the $p$ shifts of $\boldsymbol{\beta}$, then the bases…

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional…

In this paper, we consider the Wright's generalized Bessel kernel $K^{(\alpha,\theta)}(x,y)$ defined by $$\theta x^{\alpha}\int_0^1J_{\frac{\alpha+1}{\theta},\frac{1}{\theta}}(ux)J_{\alpha+1,\theta}((uy)^{\theta})u^\alpha\,\mathrm{d} u,…

We consider the problem of partitioning a line segment into two subsets, so that $n$ finite measures all have the same ratio of values for the subsets. Letting $\alpha\in[0,1]$ denote the desired ratio, this generalises the PPA-complete…

The set $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ of all finite subsets of $\mathbb{N}_0$ containing the zero element is a monoid with set addition as operation. If a set $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ can be written in the…

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$,…

A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it…