Related papers: Sets whose differences avoid squares modulo m

We prove that every transitive and non minimal semigroup with dense minimal points is sensitive. When the system is almost open, we obtain a generalization of this result.

A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x\ne y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = \Omega(n^{0.5(1+ \log_{65}…

For every nonconstant monic polynomial $g \in \mathbb{Z}[X]$, let $\mathfrak{M}(g)$ be the set of positive integers $m$ for which there exist an integer linear recurrence $(s_n)_{n \geq 0}$ having characteristic polynomial $g$ and a…

Define the map $\mathsf{T}$ on the positive integers by $\mathsf{T}(m)=\frac{m}{2}$ if $m$ is even and by $\mathsf{T}(m)=\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\epsilon>0$, almost all…

Majorana zero modes (MZMs) are non-Abelian quasiparticles with the potential to serve as topological qubits for fault-tolerant quantum computing due to their ability to encode quantum information nonlocally. In multi-Majorana systems…

A set $\mathcal{A}\subset \mathbb{N}$ is called additively decomposable (resp. asymptotically additively decomposable) if there exist sets $\mathcal{B},\mathcal{C}\subset \mathbb{N}$ of cardinality at least two each such that…

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+\beta)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results…

Motzkin posed the problem of finding the maximal density $\mu(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In…

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is…

We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing…

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha…

The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T^{N,\chi}_{l,k} for all primes l, all weights k>1 and all…

Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from…

A result of Nayak asserts that $\underset{m\to \infty}\lim |A^m|^{1/m}$ exists for each $n\times n$ complex matrix $A$, where $|A| = (A^*A)^{1/2}$, and the limit is given in terms of the spectral decomposition. We extend the result of…

In this paper we prove a theorem about the coefficients in a block of a half integral weight modular form. We show that the result of Serre and Stark for weight 1/2 forms does not generalize to higher higher weights. Let f be a half…

Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$…

The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when…

Write $\zeta_m(n)$, $1\le m\le n-1$, for the negative zeros of the $n$-th Bell polynomial, ordered in decreasing size. In this paper, we prove the following asymptotic: for a positive integer $m$ we have $$ \lim_{n\to…

Let $Z$ be a finite set of $s$ points in the projective space $\mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $\alpha(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in…

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…