Product of difference sets of set of primes

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|...

Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a prime p such that x-y=\psi(p-1). Furthermore, if P be a set of primes with the positive relative upper density, then there exist x,y\in P and a prime p such that x-y=\psi(p-1).

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any $\varepsilon>0$, we show there are sets $\mathcal{A}$ of density $>1-\varepsilon$ for which the product set $\mathcal{A}\cdot\mathcal{A}$ has density $<\varepsilon$. This answers two questions of Hegyv\'{a}ri, Hennecart and Pach.

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.

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar result takes place in the prime field F_p for sufficiently small D. It gives, in particular, that multiplicative subgroups of size less than p^{4/5-\eps} cannot be represented in the form A-A for any A from F_p.

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.

For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is $E$-intersective if for every $A \subset E$ having positive upper relative density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When $E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic intersectivity. In this article, ...

We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab=c, with a,b,c in the set. We also consider some natural generalizations, as well as a specific numerical example of a product-free set of integers with asymptotic density greater than 1/2.

We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$, and deduce asymptotic formulae with error terms regarding this problem and analogous ones. We give numerical approximations of the constants $\delta_k$ and improve the error term of formula (1.2) due to {\sc E. Cohen}. We point out that ou...

Motivated by a question of S\'ark\"ozy, we study the gaps in the product sequence $\B=\A ... \A=\{b_n=a_ia_j, a_i,a_j\in \A\}$ when $\A$ has upper Banach density $\alpha>0$. We prove that there are infinitely many gaps $b_{n+1}-b_n\ll \alpha^{-3}$ and that for $t\ge2$ there are infinitely many $t$-gaps $b_{n+t}-b_{n}\ll t^2\alpha^{-4}$. Furthermore we prove that these estimates are best possible. We also discuss a related question about the cardinality of the quotient set $...