On some applications of GCD sums to arithmetic combinatorics

JOURNAL OF NUMBER THEORY

卷:231

Article

Shkredov, Ilya D.¹ 

[1] Steklov Math Inst, Ul Gubkina 8, Moscow 119991, Russia

关键词: GCD sums;    Arithmetic combinatorics;    Sum-product;   

DOI  :  10.1016/j.jnt.2021.05.011

来源: Elsevier

【 摘 要 】

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 vertical bar S vertical bar . log log vertical bar S vertical bar). 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. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free