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.

翻译：我们证明了关于有界均匀分布与小偏差分布的汉明权重的若干新结果。我们构造了具有反集中性质的汉明权重有界均匀分布，匹配了现有的集中不等式。该构造依赖于Erd\'eyi（Acta Arithmetica 2016）在逼近论中的最新结果。特别地，我们匹配了经典的尾界，推广了Bun与Steinke（RANDOM 2015）的结果。同时，我们改进了Benjamini、Gurel-Gurevich与Peled（2012）的构造。我们提出了一种通用变换，可将任意有界均匀分布转换为小偏差分布，并几乎保持其权重分布。结合上述结果与其他方法应用该变换，我们构造了具有多种权重限制的小偏差分布。特别地，我们匹配了由有界均匀分布导出的集中性以及小偏差分布与有界均匀分布的通用接近性，回答了Bun与Steinke（RANDOM 2015）提出的问题。此外，这些分布仅支撑在常数个汉明权重上。我们进一步将反集中构造推广到受噪声扰动的小偏差分布，此类分布在近期去随机化研究中备受关注。我们的结果蕴含（但不被蕴含于）作者近期的一项成果（CCC 2024），且基于不同的技术方法。特别地，我们证明了标准高斯分布远离任何有界方差的高斯混合分布。