We prove that if \(B\) is a set of \(N\) positive integers such that \(B\cdot B\) contains an arithmetic progression of length \(M\) then \(N\geq \pi(M) + M^{2/3-o(1)}\). On the other hand, there are examples for which \(N< \pi(M)+ M^{2/3}\). This improves previously known bounds of the form \(N = \Omega(\pi(M))\) and \(N = O(\pi(M))\), respectively. The main new tool is a reduction of the original problem to the question of an approximate additive decomposition of the \(3\)-sphere in \(\mathbb{F}_3^n\) which is the set of 0-1 vectors with exactly three non-zero coordinates. Namely, we prove that such a set cannot be contained in a sumset \(A+A\) unless \(|A| \gg n^2\).