Improved bounds for arithmetic progressions in product sets

Abstract

Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B=\{b{b}^{\prime }|b,b^{\prime }\in B\}$ cannot be greater than $O(n\log n)$ which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers we improve the bound to $O_{\epsilon }(n^{1+\epsilon })$ for arbitrary $\epsilon >0$ assuming the GRH.

To appear in Int. J. Number Theory