Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the size of subsets of $\mathbb{F}_q^n$ avoiding solutions to linear
  systems with repeated columns   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 $\gamma < q$ 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.
  In this paper, we present general techniques for systems with repeated
columns. This class of linear systems is disjoint from the class covered by
Sauermann's result, and captures the systems studied by Mimura and Tokushige
into a single proof. 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 (where $p \geq k \geq 3$), then $S$ must have
exponentially small density.

27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat