Product sets cannot contain long arithmetic progressions

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 $O(\frac{n{\log }^{2}n}{\log \log n})$ and present an example of a product set containing an arithmetic progression of length $\Omega (n\log n)$. For sets of complex numbers we obtain the upper bound $O(n^{3/2})$.

arXiv commentThe previous version contained an error in the complex case. This is a corrected version with a weaker bound $n^{3/2}$ for sets of complex numbers and will appear in Acta Arithmetica