SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 730-741, January 2020.
If $k<n$, can one express the majority of $n$ bits as the majority of at most $k$ majorities, each of at most $k$ bits? We prove that such an expression is possible only if $k \gtrsim n^{4/5} = n^{0.8}$. This improves on a bound proved by Kulikov and Podolskii [Computing majority by constant depth majority circuits with low fan-in gates, in Proceedings of the 34th STACS, Schloss Dagstuhl, Wadern, Germany, 2017, 49], who showed that $k \gtrsim n^{0.7 + o(1)}$. Our proof is based on ideas originating in discrepancy theory as well as a strong concentration bound for sums of independent Bernoulli random variables and a strong anticoncentration bound for the hypergeometric distribution.

中文翻译：

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论多数为多数

SIAM离散数学杂志，第34卷，第1期，第730-741页，2020年1月。
如果$ k <n $，可以将$ n $位的大部分表示为最多$ k $多数的多数，每个最多$ k $位？我们证明只有$ k \ gtrsim n ^ {4/5} = n ^ {0.8} $时，这样的表达式才可能。这在由Kulikov和Podolskii证明的边界上得到了改善[在34th STACS的会议记录中，采用低扇入门的恒定深度多数电路计算多数，Schloss Dagstuhl，德国瓦登，2017年，49]，他们证明了$ k \ gtrsim n ^ {0.7 + o（1）} $。我们的证明基于差异理论的思想，以及独立伯努利随机变量和的强集中度以及超几何分布的强反集中度。