Counting solutions to invariant equations in dense sets

TL;DR

This paper establishes lower bounds and constructions for the number of solutions to invariant equations in dense sets, and improves bounds on the size of sets avoiding certain solutions, advancing understanding in additive combinatorics.

Contribution

It provides new lower bounds, a Behrend-type construction, and improved size bounds for sets avoiding solutions to invariant equations.

Findings

Lower bound of exp(-C (log(2/alpha))^7)N^{k-1} for solutions in dense sets

Behrend-type construction with upper bound exp(-c (log(2/alpha))^2)N^{k-1}

Sets avoiding certain solutions have size at most exp(-c(log N)^{1/(6+gamma)})N

Abstract

We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the number of solutions of a convex equation bounded above by exp(-c (log(2/alpha))^2)N^{k-1}. Furthermore, improving the result of Schoen and Sisask, we show that if a set does not contain any non-trivial solutions to an equation of length at least 2(3^{m+1})+2 for some positive integer m, then its size is at most exp(-c(log N)^{1/(6+gamma)})N, where gamma = 2^{1-m}.