= Product mixing in a quasirandom group
{title2=$\|f*g\|_2\leq m^{-1/2}\|f\|_2\|g\|_2$}
For an $m$-<quasirandom group>, uniform <expectations>, and the <normalized convolution on a finite group>, a scalar <mean-zero function> $f$ satisfies
$$
\|f*g\|_2\leq m^{-1/2}\|f\|_2\|g\|_2.
$$
The <Fourier analysis on a finite group> proof bounds each nontrivial matrix component of $f$ in <operator norm> using its weighted <Hilbert-Schmidt norm>. For subsets of <subset density> values $a,b,c$, the error in their normalized product count is at most $\sqrt{a(1-a)b(1-b)c(1-c)/m}$. In particular $abc>1/m$ guarantees a solution of $xy=z$ in the three subsets.
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