Minimizing the frequency of carries in modular addition

TL;DR

This paper proves that in modular addition with digital sets, carries occur at least 25% of the time asymptotically, extending previous results to a broader setting.

Contribution

It generalizes prior work by showing a lower bound on carry frequency for digital sets in modular addition for any divisor of q.

Findings

Carries occur at least 1/4 of the time asymptotically.

The result generalizes previous bounds for specific cases.

Provides a new understanding of carry behavior in modular arithmetic.

Abstract

When adding integers in base $m$, carries occur. The same happens modulo a generic integer $q$ when the set of digits is a complete set of residues modulo $m$ for some positive integer $m$ dividing $q$. In this paper we prove that asymptotically every digital set in this setting induces carries with frequency at least $1/4$, thus generalizing results of Alon, Diaconis, Shao and Soundararajan.