Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences

TL;DR

This paper investigates the periodicity properties of exponential sums of symmetric Boolean functions modulo primes, establishing bounds and introducing the concept of avoiding primes to determine non-balanced functions.

Contribution

It introduces bounds on periods of exponential sums, the concept of avoiding primes, and applies these to show certain symmetric Boolean functions are not balanced.

Findings

Elementary symmetric Boolean functions of degree not a power of 2 are not balanced.

Certain primes avoid the exponential sums of specific symmetric Boolean functions.

The set of primes dividing some exponential sum terms has positive density.

Abstract

This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these exponential sums. The concept of avoiding primes is also introduced. This concept and the bounds presented in this work are used to show that some classes of symmetric Boolean functions are not balanced. In particular, every elementary symmetric Boolean function of degree not a power of 2 and less than 2048 is not balanced. For instance, the elementary symmetric Boolean function in $n$ variables of degree $1292$ is not balanced because the prime $p = 176129$ does not divide its exponential sum for any positive integer $n$. It is showed that for some symmetric Boolean functions, the set of primes avoided by the sequence of exponential sums contains a subset that has positive density within the set of primes. Finally, in the last section, a brief study for the set of primes that divide some term of the sequence of exponential sums is presented.