{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:50Z","timestamp":1753893770050,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Consider a system of $m$ balanced linear equations in $k$ variables with coefficients in $\\mathbb{F}_q$. If $k \\geq 2m + 1$, then a routine application of the slice rank method shows that there are constants $\\beta,\\gamma \\geq 1$ with $\\gamma &lt; q$ such that, for every subset $S \\subseteq \\mathbb{F}_q^n$ of size at least $\\beta \\cdot \\gamma^n$, the system has a solution $(x_1,\\ldots,x_k) \\in S^k$ with $x_1,\\ldots,x_k$ not all equal. Building on a series of papers by Mimura and Tokushige and on a paper by Sauermann, this paper investigates the problem of finding a solution of higher non-degeneracy; that is, a solution where $x_1,\\ldots,x_k$ are pairwise distinct, or even a solution where $x_1,\\ldots,x_k$ do not satisfy any balanced linear equation that is not a linear combination of the equations in the system.\r\nIn this paper, we focus on linear systems with repeated columns. For a large class of systems of this type, we prove that there are constants $\\beta,\\gamma \\geq 1$ with $\\gamma &lt; q$ such that every subset $S \\subseteq \\mathbb{F}_q^n$ of size at least $\\beta \\cdot \\gamma^n$ contains a solution that is non-degenerate (in one of the two senses described above). This class is disjoint from the class covered by Sauermann's result, and captures the systems studied by Mimura and Tokushige into a single proof. Moreover, a special case of our results shows that, if $S \\subseteq \\mathbb{F}_p^n$ is a subset such that $S - S$ does not contain a non-trivial $k$-term arithmetic progression (with $p$ prime and $3 \\leq k \\leq p$), then $S$ must have exponentially small density.<\/jats:p>","DOI":"10.37236\/10883","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T09:55:19Z","timestamp":1696586119000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Size of Subsets of $\\mathbb{F}_q^n$ Avoiding Solutions to Linear Systems with Repeated Columns"],"prefix":"10.37236","volume":"30","author":[{"given":"Josse","family":"Van Dobben de Bruyn","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dion","family":"Gijswijt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2023,10,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i4p1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i4p1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T09:55:19Z","timestamp":1696586119000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i4p1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,6]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,10,6]]}},"URL":"https:\/\/doi.org\/10.37236\/10883","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,10,6]]},"article-number":"P4.1"}}