Constructive Relationships Between Algebraic Thickness and Normality

TL;DR

This paper explores the connection between algebraic thickness and normality in Boolean functions, establishing bounds on affine subspaces where functions are constant, and analyzing the algebraic thickness of the majority function.

Contribution

It provides new bounds linking algebraic thickness and normality, and introduces an algorithm to find affine subspaces where functions are constant, advancing understanding of Boolean function structure.

Findings

Functions with algebraic thickness $n^{3- ext{epsilon}}$ are constant on large affine subspaces.

Majority function has exponential algebraic thickness of $ ext{Omega}(2^{n^{1/6}})$.

The proposed algorithm approximates the best affine subspace with a factor close to $ ext{sqrt}(n)$.

Abstract

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < \epsilon<2$, any function with algebraic thickness $n^{3-\epsilon}$ is constant on some affine subspace of dimension $\Omega\left(n^{\frac{\epsilon}{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $\Theta(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $\Theta(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $\Omega\left(2^{n^{1/6}}\right)$.