Difference sets are not multiplicatively closed

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|...

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set $A$ of positive real numbers, it is true that $$\left|\left\{\frac{a+b}{c+d}:a,b,c,d\in{A}\right\}\right|\geq{2|A|^2-1}.$$ As a consequence of this result, it is also established that $$|4^{k-1}A^{(k)}|:=|\underbrace{\underbrace{A\cdots{A...

In a recent work \cite{key-11}, A. Fish proved that if $E_{1}$ and $E_{2}$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in\mathbb{Z}$ such that $k\cdot\mathbb{Z}\subset\left(E_{1}-E_{1}\right)\cdot\left(E_{2}-E_{2}\right).$ In this article we will show that a similar result is true for the set of primes $\mathbb{P}$ (which has density $0$). We will prove that there exists $k\in\mathbb{N}$ such that $k\cdot\mathbb{N}\subset\left(\mathb...

We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and ...

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.

We show two asymmetric estimates, one on the number of collinear triples and the other on that of solutions to $(a_1+a_2)(a_1^{\prime\prime\prime}+a_2^{\prime\prime\prime})=(a_1^\prime+a_2^\prime)(a_1^{\prime\prime}+a_2^{\prime\prime})$. As applications, we improve results on difference-product/division estimates and on Balog-Wooley decomposition: For any finite subset $A$ of $\mathbb{R}$, \[ \max\{|A-A|,|AA|\} \gtrsim |A|^{1+105/347},\quad \max\{|A-A|,|A/A|\} \gtrsim |A|^{1+...

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.

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size $k$ satisfies $|A'-A'|\ge k^{\log_2 3}$. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties.

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if $|A|\succeq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succapprox \min\{|A|^{13/12}(\frac{|A|}{p^{0.5}})^{1/12},|A|(\frac{p}{|A|})^{1/11}\}.\] These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on differ...

New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising his approach to the complex plane. The bounds involving the difference set are slightly weaker. They improve on the best known ones, including the case $A\subset \R$, which also due to Solymosi, by means of combining the use of the Szemer\'e...