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 ... more >>>

We prove several new results on the Hamming weight of bounded uniform and small-bias distributions.

We exhibit bounded-uniform distributions whose weight is anti-concentrated, matching existing concentration inequalities. This construction relies on a recent result in approximation theory due to Erdéyi (Acta Arithmetica 2016). In particular, we match the classical tail ... more >>>

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that

... more >>>