Product sets cannot contain long arithmetic progressions
Volume 163 / 2014
Acta Arithmetica 163 (2014), 299-307
MSC: Primary 11B25.
DOI: 10.4064/aa163-4-1
Abstract
Let $B$ be a set of complex numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'\mid b, b' \in B\}$ 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 $O(n^{3/2})$.
