Covering almost all the layers of the hypercube with multiplicities

Abstract

Given a hypercube Qn:={0,1}n in Rn and k∈{0,…,n}, the k-th layer Qkn of Qn denotes the set of all points in Qn whose coordinates contain exactly k many ones. For a fixed t∈N and k∈{0,…,n}, let P∈R[x1,…,xn] be a polynomial that has zeroes of multiplicity at least t at all points of Qn∖Qkn, and P has zeros of multiplicity exactly t−1 at all points of Qkn. In this short note, we show thatdeg(P)≥max⁡{k,n−k}+2t−2. Matching the above lower bound we give an explicit construction of a family of hyperplanes H1,…,Hm in Rn, where m=max⁡{k,n−k}+2t−2, such that every point of Qkn will be covered exactly t−1 times, and every other point of Qn will be covered at least t times. Note that putting k=0 and t=1, we recover the much celebrated covering result of Alon and Füredi (1993) [1]. Using the above family of hyperplanes we disprove a conjecture of Venkitesh (2022) [23] on exactly covering symmetric subsets of hypercube Qn with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest.We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.