Sets whose differences avoid squares modulo m

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively, non-residues) of p}. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and variations of this a...

Let $h$ be a positive integer and let $\varepsilon > 0$. The Haight-Ruzsa method produces a positive integer $m^*$ and a subset $A$ of the additive abelian group $\mathbf{Z}/m^*\mathbf{Z}$ such that the difference set is large in the sense that $A-A = \mathbf{Z}/m^*\mathbf{Z}$ and $h$-fold sumset is small in the sense that $|hA| < \varepsilon m^*$. This note describes, and in a modest way extends, the Haight-Ruzsa argument, and constructs sets with more differences than multi...

We show that if A is a subset of {1, ..., n} such that it has no pairs of elements whose difference is equal to p-1 with p a prime number, then the size of A is O(n(loglog n)^(-clogloglogloglog n)) for some positive constant c.

For $\ell\geq 2$ and $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ of degree $k\geq 2$, we show that every set $A\subseteq \{1,2,\dots,N\}$ lacking nonzero differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_h Ne^{-c(\log N)^{\mu}}$, where $c=c(h)>0$, $\mu=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $\mu=1/2$ if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple of every natural number and $h$ satisfies certain nonsingularity conditions. We also explore these conditions in ...

We prove, in particular, that for any finite set of real numbers A with |A/A| \ll |A| one has |A-A| > |A|^{5/3 - o(1)}. Also we show that |3A| > |A|^{2-o(1)} in the case.

In our paper, we apply additive-combinatorial methods to study the distribution of the set of squares $\mathcal{R}$ in the prime field. We obtain the best upper bound on the number of gaps in $\mathcal{R}$ at the moment and generalize this result for sets with small doubling.

We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies $\Delta(h)=b^2-4ac\neq 0$, then any subset of $\{1,2,\dots,N\}$ lacking nonzero differences in the image of $h$ has size at most a constant depending on $h$ times $N\exp(-c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an $L^2$ density increment strategy previously used to establish analogous results for sums of one or more single-variable polynomials. Our exposition is thorough and self-...

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N)^{-c\log\log\log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent r...

In this paper we prove that given two sets $E_1,E_2 \subset \mathbb{Z}$ of positive density, there exists $k \geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $\alpha,\beta > 0$ then there exist $N_0$ and $d_0$ which depend only on $\alpha$ and $\beta$ such that for every $N \geq N_0$ and $E_1,E_2 \subset \mathbb{Z}_N$ with $|E_1|...

We show that the number of positive integers $n\leq N$ such that $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ contains a perfect difference set is asymptotically $N/\log{N}$.