  Using GCD sums, we show that the set of the primes has small common
multiplicative energy with an arbitrary exponentially big integer set $S$ and,
in particular, size of any arithmetic progression in $S$ having the beginning
at zero, is at most $O(\log |S| \cdot \log \log |S|)$. This result can be
considered as an integer analogue of Vinogradov's question about the least
quadratic non--residue. The proof rests on a certain repulsion property of the
function $f(x)=\log x$. Also, we consider the case of general $k$--convex
functions $f$ and obtain a new incidence result for collections of the curves
$y=f(x)+c$.
