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arXiv · 2408.10881

Sets with no solutions to some symmetric linear equations

Tomasz Kosciuszko
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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.

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Tomasz Kosciuszko. 2024-08-20. Sets with no solutions to some symmetric linear equations. https://arxiv.org/abs/2408.10881

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