## Abstract

Consider a system of m balanced linear equations in k variables with coefficients in F_{q}. If k ⩾ 2m + 1, then a routine application of the slice rank method shows that there are constants β, γ ⩾ 1 with γ < q such that, for every subset S ⊆ F^{n}q of size at least β · γ^{n}, the system has a solution (x_{1}, …, x_{k}) ∈ S^{k} with x_{1}, …, 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}, …, x_{k} are pairwise distinct, or even a solution where x_{1}, …, x_{k} do not satisfy any balanced linear equation that is not a linear combination of the equations in the system. In 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 β, γ ⩾ 1 with γ < q such that every subset S ⊆ F^{n}q of size at least β · γ^{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 ⊆ F^{n}p is a subset such that S − S does not contain a non-trivial k-term arithmetic progression (with p prime and 3 ⩽ k ⩽ p), then S must have exponentially small density.

Original language | English |
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Number of pages | 30 |

Journal | Electronic Journal of Combinatorics |

Volume | 30 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2023 |

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