Given a distribution over $[n]^n$ such that any $k$ coordinates need $k/\log^{O(1)}n$ bits of communication to sample, we prove that any map that samples this distribution from uniform cells requires locality $\Omega(\log(n/k)/\log\log(n/k))$. In particular, we show that for any constant $\delta>0$, there exists $\varepsilon=2^{-\Omega(n^{1-\delta})}$ such that $\Omega(\log n/\log\log n)$ non-adaptive ... more >>>

We consider the following problem: Given an $n \times n$ multiplication table, decide whether it is a Cayley multiplication table of a group. Among deterministic algorithms for this problem, the best known algorithm is implied by F. W. Light's associativity test (1949) and has running time of $O(n^2 \log n)$. ... more >>>

We define the marginal information of a communication protocol, and use it to prove XOR lemmas for communication complexity. We show that if every $C$-bit protocol has bounded advantage for computing a Boolean function $f$, then every $\tilde \Omega(C \sqrt{n})$-bit protocol has advantage $\exp(-\Omega(n))$ for computing the $n$-fold xor $f^{\oplus ... more >>>