Pseudorandomness, symmetry, smoothing: II

Abstract: 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 Erd\'eyi (Acta Arithmetica 2016). In particular, we match the classical tail bounds, generalizing a result by Bun and Steinke (RANDOM 2015). Also, we improve on a construction by Benjamini, Gurel-Gurevich, and Peled (2012). We give a generic transformation that converts any bounded uniform distribution to a small-bias distribution that almost preserves its weight distribution. Applying this transformation in conjunction with the above results and others, we construct small-bias distributions with various weight restrictions. In particular, we match the concentration that follows from that of bounded uniformity and the generic closeness of small-bias and bounded-uniform distributions, answering a question by Bun and Steinke (RANDOM 2015). Moreover, these distributions are supported on only a constant number of Hamming weights. We further extend the anti-concentration constructions to small-bias distributions perturbed with noise, a class that has received much attention recently in derandomization. Our results imply (but are not implied by) a recent result of the authors (CCC 2024), and are based on different techniques. In particular, we prove that the standard Gaussian distribution is far from any mixture of Gaussians with bounded variance.