A Polynomial Bogolyubov type Result for the Special Linear Group
summary
Result: for A ⊂ SLn(Fq) of density as small as q^{-c n^2}, the product A A^{-1} A A^{-1} contains a subgroup whose relative density is polynomial in the density of A rather than quasipolynomial. Proof combines a new hyper-contractive estimate (inspired by 2-to-2 games), convolution/representation analysis showing f∗g is nearly constant, lower bounds on irreducible dimensions via tensor rank, and a level-d/Bonami-type inequality for global sets; the argument yields explicit rank and exponent bounds.
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