Minimizing the frequency of carries in modular addition

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.