ON EXPRESSING MAJORITY AS A MAJORITY OF MAJORITIES

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 greater than or similar to 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 greater than or similar to 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.