Mixing of 3-term progressions in Quasirandom Groups
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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 ⊂ G, we have |_x,y∼ G[ x ∈ A_1, xy ∈ A_2, xy^2 ∈ A_3] - ∏_i=1^3 _x∼ G[x ∈ A_i] | ≤(2/√(D))^1/4. Prior to this, Tao answered this question when the underlying quasirandom group is SL_d(𝔽_q). Subsequently, Peluse extended the result to all nonabelian finite 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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