Product sets cannot contain long arithmetic progressions

Journal article, 2014

Let B be a set of real numbers of size n . We prove that the length of the longest arithmetic progression contained in the product set B.B={bibj|bi,bj∈B}B.B={bibj|bi,bj∈B} cannot be greater than View the MathML sourceO(n1+1/loglogn) an arithmetic progression of length View the MathML sourceΩ(nlogn), so the obtained upper bound is close to the optimal.