Improved bounds for arithmetic progressions in product sets

TL;DR

This paper establishes upper bounds on the length of arithmetic progressions within product sets of natural numbers and complex numbers, improving previous bounds and matching known lower bounds up to constants.

Contribution

It provides new bounds for arithmetic progressions in product sets of natural and complex numbers, assuming GRH for the complex case.

Findings

Longest AP in natural number product sets is O(n log n)

Bound for complex numbers is O_ε(n^{1+ε}) under GRH

Results match lower bounds up to constants

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 $B.B = \{bb'| \, b, b' \in B\}$ 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.