Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mixing of 3-term progressions in Quasirandom Groups   In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$
in every finite quasirandom groups, fully answering a question of Gowers. More
precisely, we show that for any $D$-quasirandom group $G$ and any three sets
$A_1, A_2, A_3 \subset G$, we have \[ \left|\Pr_{x,y\sim G}\left[ x \in A_1, xy
\in A_2, xy^2 \in A_3\right] - \prod_{i=1}^3 \Pr_{x\sim G}\left[x \in
A_i\right] \right| \leq \left(\frac{2}{\sqrt{D}}\right)^{\frac{1}{4}}.\] Prior
to this, Tao answered this question when the underlying quasirandom group is
$\mathrm{SL}_{d}(\mathbb{F}_q)$. Subsequently, Peluse extended the result to
all nonabelian finite $\textit{simple}$ groups. In this work, we show that a
slight modification of Peluse's argument is sufficient to fully resolve Gower's
quasirandom conjecture for 3-term progressions. Surprisingly, unlike the proofs
of Tao and Peluse, our proof is elementary and only uses basic facts from
nonabelian Fourier analysis.

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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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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: stat
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The categories of this paper are: ao-sci