Sets whose differences avoid squares modulo m

We prove that for all squarefree $m$ and any set $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero squares the bound $|A|\leq m^{1/2}(3n)^{1.5n}$ holds, where $n$ denotes the number of odd prime divisors of $m$.

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$.

We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemer\'edi.

In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq \mathbb{Z}/q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

Let $S$ be a subset of $\{1,\ldots,N\}$ avoiding the nontrivial progressions $x, x+y^2-1, x+ 2(y^2-1)$. We prove that $|S|\ll N/\log_m{N}$, where $\log_m $ is the $m$-fold iterated logarithm and $m\in\mathbf{N}$ is an absolute constant. This answers a question of Green.

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. W...

Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.

Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]\cap\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1...

Let $A\subset [1,N]$ be a set of positive integers with $|A|\gg \sqrt N$. We show that if avoids about $p/2$ residue classes modulo $p$ for each prime $p$, the $A$ must correlate additively with the squares $S=\{n^2:1\leq n\leq \sqrt N\}$, in the sense that we have the additive energy estimate $E(A,S)\gg N\log N$. This is, in a sense, optimal.

Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a subset $U$ of $\mathbb{Z}/M\mathbb{Z}$ to be the sum of squares of the consecutive distances between elements of $U$. In this paper we study the stochasticity parameter of the set $R_M$ of quadratic residues modulo $M$. We present a method which allows to find the asymptotics of $S(R_M)$ for a set of $M$ of positive density. In particular, we obtain the following two corollaries. Denote by $s(k)=s(k,\ma...