On a question of Sárközy on gaps of product sequences

Abstract

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b ₁ < b ₂ < …} of all products a _i a _j with a _i , a _j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b _n+1 − b _n ≪ α ⁻³ and that for t ≥ 2 there are infinitely many t-gaps b _n+t − b _n ≪ t ² α ⁻⁴. Furthermore, we prove that these estimates are best possible.

We also discuss a related question about the cardinality of the quotient set A/A = {a _i /a _j , a _i , a _j ∈ A} when A ⊂ {1, …, N} and |A| = αN.