On some applications of GCD sums to Arithmetic Combinatorics

TL;DR

This paper applies GCD sums to arithmetic combinatorics, demonstrating small multiplicative energy between primes and large sets, and establishing bounds on arithmetic progressions, with implications for quadratic non-residues and incidence geometry.

Contribution

It introduces new bounds on the intersection of prime sets with large sets using GCD sums and extends incidence results to convex functions.

Findings

Primes have small multiplicative energy with large sets.

Bound on size of arithmetic progressions starting at zero in large sets.

New incidence results for curves defined by convex functions.

Abstract

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 $y=f(x)+c$.