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

On some applications of GCD sums to Arithmetic Combinatorics