Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially large integer set S. This implies that if S is a set of small multiplicative doubling then the size of any arithmetic progression in S, 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 $y=f(x)+c$.

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关于 GCD 和在算术组合中的一些应用

使用 GCD 和，我们证明素数集具有小的公共乘法能量和任意指数大整数集S。这意味着如果S是一组小的乘法加倍，那么S中任何等差数列的大小，从零开始，至多是$哦(\mathrm{日志}|秒|\cdot \mathrm{日志}\mathrm{日志}|秒|)$. 这个结果可以被认为是维诺格拉多夫关于最小二次非残差问题的整数模拟。证明依赖于函数的某种排斥性$F(X)=\mathrm{日志}X$. 此外，我们考虑一般k-凸函数f的情况，并获得曲线集合的新关联结果$是=F(X)+C$.