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<p>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 is 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 equations in 6 variables. In the final section we present a couple of remarks on non-symmetric equations.</p>

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Sets with no solutions to some symmetric linear equations

  • 期刊论文

  • 发表日期：

    2025年4月21日

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  • Tomasz Kościuszko

摘要

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 is 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 equations in 6 variables. In the final section we present a couple of remarks on non-symmetric equations.