Boosting uniformity in quasirandom groups: fast and simple
<|reference_start|>Boosting uniformity in quasirandom groups: fast and simple: We study the communication complexity of multiplying $k\times t$ elements from the group $H=\text{SL}(2,q)$ 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 $H$, 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.<|reference_end|>
@article{derksen2024boosting,
    title={Boosting uniformity in quasirandom groups: fast and simple},
    author={Harm Derksen, Chin Ho Lee, Emanuele Viola},
    journal={arXiv preprint arXiv:2409.06932},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.06932},
    primaryClass={cs.CC math.CO}
}