Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sets avoiding $p$-term arithmetic progressions in ${\mathbb Z}_{q}^n$
  are exponentially small   Pach and Palincza proved the following generalization of Ellenberg and
Gijswijt's bound for the size of $k$-term arithmetic progression-free subsets,
where $k\in \{4,5,6\}$:
  Let $m>0$ be an integer such that $6$ divides $m$ and let $k\in \{4,5,6\}$.
Then $$ r_k({\mathbb Z}_{m}^n)\leq (0.948m)^n $$ if $n$ is sufficiently large.
  Building on the proof technique of Pach and Palincza's upper bound we
generalize the Ellenberg and Gijswijt's bound in the following way:
  Let $p>2$ be any integer and let $q>2$ be a prime. Suppose that $p\leq q$.
Then the there exists an $n_0\in \mathbb N$ integer and a $0<\delta(p,q)<1$
real number such that $$ r_p({\mathbb Z}_{q}^n)\leq (\delta(p,q)q)^n $$ for
each $n>n_0$.

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The categories of this paper are: nlin
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The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: cs
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn