We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map $f : \mathbb{F}_q^r \to \mathbb{F}_q^n$ to an element sampled uniformly at random from a $k$-dimensional variety $V \subseteq \mathbb{F}_q^r$. This class of sources generalizes both polynomial sources, studied by Dvir, ... more >>>

Abstract The Union Closed Set Conjecture states that if a set system X\subseteq\mathcal{P}([n]) is closed under pairwise unions, then there exists a\in[n] in at least half of the sets of X. We show that there is a very natural generalization of the union closed set conjecture which gives a lower ... more >>>

We show that the value of the $n$-fold repeated GHZ game is at most $2^{-\Omega(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.

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