  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 $C >
0$,
  $$
  \pi(M) + C \frac {M^{2/3}}{\log^2 M} \leq N,
  $$ where $\pi$ is the prime counting function.
  This improves on previously known bounds of the form $N = \Omega(\pi(M))$ and
gives a bound which is sharp up to the second order term, as Pach and S\'andor
gave an example for which
  $$
  N < \pi(M)+ O\left(\frac {M^{2/3}}{\log^2 M} \right).
  $$
  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 $c > 0$.
