论文标题
在GCD总和到算术组合的某些应用中
On some applications of GCD sums to Arithmetic Combinatorics
论文作者
论文摘要
使用GCD总和，我们表明，素数的集合具有少量的乘数能量，具有任意指数的大整数集$ s $，尤其是在$ s $中以零开始的任何算术进程的大小，最多是$ o（\ log log | s | s | \ cdot \ cdot \ log \ log \ log \ log \ log | s |）$。该结果可以被视为Vinogradov关于最小二次非放弃的问题的整数类似物。证明基于函数$ f（x）= \ log x $的特定排斥属性。此外，我们考虑了一般$ k $ - 凸函数$ f $的情况，并获得了曲线收集$ y = f（x）+c $的新发病率结果。
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 $y=f(x)+c$.
