Covering the hypercube, the uncertainty principle, and an interpolation formula

TL;DR

The minimal covering problems are closely related to uncertainty principle on the hypercube, and an interpolation formula is obtained for multilinear polynomials on $\mathbb{R}^{n}$ of degree less than $\lfloor n/m \rfloor$ by showing that its coefficients corresponding to the largest monomials can be represented as a linear combination of values of the polynomial over the points.

Abstract

We show that the minimal number of skewed hyperplanes that cover the hypercube $\{0,1\}^{n}$ is at least $\frac{n}{2}+1$, and there are infinitely many $n$'s when the hypercube can be covered with $n-\log_{2}(n)+1$ skewed hyperplanes. The minimal covering problems are closely related to uncertainty principle on the hypercube, where we also obtain an interpolation formula for multilinear polynomials on $\mathbb{R}^{n}$ of degree less than $\lfloor n/m \rfloor$ by showing that its coefficients corresponding to the largest monomials can be represented as a linear combination of values of the polynomial over the points $\{0,1\}^{n}$ whose hamming weights are divisible by $m$.

Theorems & Definitions (14)

• Proposition 1
• Theorem 3
• Theorem 4: Linial--Radhakrishnan Linial
• Definition 5
• Theorem 6
• Corollary 7
• Remark 8
• Remark 9
• Corollary 10
• Lemma 11