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 >>>

We revisit the problem of constructing explicit pseudorandom generators
that fool with error $\epsilon$ degree-$d$ polynomials in $n$ variables
over the field $F_q$, in the case of large $q$. Previous constructions
either have seed length at least $2^{d}\log q$, and thus are only non-trivial
when the degree is less than ... more >>>

Let $G$ be a group such that any non-trivial representation has dimension
at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$
be distributions over $G^{t}$. Suppose that $X$ is independent from
$Y$. We show that for any $g\in G$ we have
\[
\left|\mathbb{P}[X_{1}Y_{1}X_{2}Y_{2}\cdots X_{t}Y_{t}=g]-1/|G|\right|\le\frac{|G|^{2t-1}}{d^{t-1}}\sqrt{\mathbb{E}_{h\in G^{t}}X(h)^{2}}\sqrt{\mathbb{E}_{h\in G^{t}}Y(h)^{2}}.
\]
Our results generalize, improve, and ... more >>>