On a Question of Sárközy on Gaps of Product Sequences

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b1 < b2 < · · · } of all products aiaj with ai, aj ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps bn+1 − bn ¿ α−3 and that for t ≥ 2 there are infinitely many t-gaps bn+t − bn ¿ t2α−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 = {ai/aj , ai, aj ∈ A} when A ⊂ {1, . . . , N} and |A| = αN .