On some applications of GCD sums to Arithmetic Combinatorics

2020/10/06 by Ilya D. Shkredov, Shkredov, Ilya D.
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.2010.03020

15 pages

arxiv created 2020/10/24 · arxiv updated 2020/10/27

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| ⋅ 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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