Boosting Uniformity in Quasirandom Groups: Fast and Simple

We study the communication complexity of multiplying $k\times t$ elements from the group $H$ = SL $(2, q)$ in the number-on-forehead model with $k$ parties. We prove a lower bound of $(t\log H)/c^{k}$. This is an exponential improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of $k^{c}$ independent copies of a 3-uniform distribution over $H^{m}$ is close to a $k$- uniform distribution. This is again an exponential improvement over previous work which needed $c^{k}$ copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group $LI$, any distribution over $H^{m}$ whose weight-k Fourier coefficients are small is close to a k-uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.