Sparse Covers for Sums of Indicators

For all $n, \epsilon >0$, we show that the set of Poisson Binomial distributions on $n$ variables admits a proper $\epsilon$-cover in total variation distance of size $n²⁺ⁿ \cdot (1/\epsilon)^{O(\log2 (1/\epsilon))}$, which can also be computed in polynomial time. We discuss the implications of our construction for approximation algorithms and the computation of approximate Nash equilibria in anonymous games.