Sets with no solutions to some symmetric linear equations
Number Theory
2024-08-21 v1 Combinatorics
Abstract
We expand the class of linear symmetric equations for which large sets with no non-trivial solutions are known. Our idea is based on first finding a small set with no solutions and then enlarging it to arbitrary size using a multi-dimensional construction, crucially assuming the equation in primitive. We start by presenting the technique on some new equations. Then we use it to show that a symmetric equation with randomly chosen coefficients has a near-optimal set with no non-trivial solutions. We also show a construction for a wide class of symmetric equation in 6 variables. In the final section we present a couple of remarks on non-symmetric equations.
Cite
@article{arxiv.2408.10881,
title = {Sets with no solutions to some symmetric linear equations},
author = {Tomasz Kosciuszko},
journal= {arXiv preprint arXiv:2408.10881},
year = {2024}
}