Counting solutions to invariant equations in dense sets
Number Theory
2023-06-16 v1 Combinatorics
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}.
Cite
@article{arxiv.2306.08567,
title = {Counting solutions to invariant equations in dense sets},
author = {Tomasz Kosciuszko},
journal= {arXiv preprint arXiv:2306.08567},
year = {2023}
}