The line--versus--point test asks the following local-to-global question. Suppose a function $f\colon\mathbb{F}_q^m\to\mathbb{F}_q$ is, on average over a random affine line $L$, correlated with some degree-$d$ polynomial on $L$. Must $f$ then be globally correlated with a single multivariate polynomial of degree at most $d$? Beyond being a natural combinatorial question, ... more >>>

A secret-sharing scheme allows a dealer to distribute a secret $s$ among $n$ parties such that only predefined “authorized” sets of parties can reconstruct the secret, and all other “unauthorized” sets learn nothing about $s$. Families of authorized sets are called access structures, and a scheme is called linear if ... more >>>

We prove a general upper bound for scaling windows of sparse monotone covering problems. From that, we deduce that for every fixed value $k\geq 3$, the window of random $k$-SAT is $O(n/\log n)$, improving the Friedgut-Bourgain bound of $O(n/\log\log n)$. We also show that random signed Not-All-Equal-$k$-SAT and hypergraph non-two-colourability ... more >>>