Product sets cannot contain long arithmetic progressions

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 = { b i b j | b i , b j ∈ B } cannot be greater than O ( n 1 + 1 / log log n ) an arithmetic progression of length Ω ( n log n ) , so the obtained upper bound is close to the optimal.