Discrete spheres and arithmetic progressions in product sets

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 for some absolute $C > 0$,$$\pi (M)+C\frac{M^{2\mathrm{/}3}}{{\log }^{2}M}\le N,$$where$\pi$ is the prime counting function. This improves on previously known bounds of the form $N=\mathrm{\Omega }(\pi (M))$ and gives a bound which is sharp up to the second order term, as Pach and S'andor gave an example for which $$N<\pi (M)+O\left(\frac{M^{2\mathrm{/}3}}{{\log }^{2}M}\right)\mathrm{.}$$ The main new tool is a reduction of the original problem to the question of 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 have an additive basis of order two of size less than $c{n}^2$ with absolute constant $c>0$.