Product sets cannot contain long arithmetic progressions
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’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound .