Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions

TL;DR

This paper improves the understanding of the linear recurrence relations and asymptotic behavior of exponential sums of symmetric Boolean functions, providing precise formulas and conditions for their balancedness.

Contribution

It offers a tighter bound on the degree of the recurrence and a formula to determine asymptotic balancedness of symmetric Boolean functions.

Findings

The degree of the recurrence is improved and shown to be tight.

A formula for asymptotic behavior of symmetric Boolean functions is provided.

Exponential sums are significantly smaller than 2^n when the degree is a power of two.

Abstract

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.