Carries and the arithmetic progression structure of sets

TL;DR

This paper investigates the structure of digit sets in base representations that minimize carry operations, generalizing previous results for prime moduli to composite moduli and exploring their connection to arithmetic progression representations.

Contribution

It extends the characterization of sets minimizing carries from prime to general moduli and links this to the uniqueness of representing sets as unions of arithmetic progressions.

Findings

Sets minimizing carries are arithmetic progressions for prime moduli.

Generalization to composite moduli shows similar structure constraints.

Connection established between minimal carry sets and unique arithmetic progression decompositions.

Abstract

If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 + a_2 \equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\mathbb{Z}/m \mathbb{Z} \subseteq \mathbb{Z}/m^2\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis, Shao and Soundararajan proved that, when $m=p$, $p$ prime, the only set $A$ which induces two distinct carries, i. e. with $A+A \subseteq \{ x, y \}+A$ for some $x, y \in \mathbb{Z}/p^2\mathbb{Z}$, is the arithmetic progression $[0, p-1]$, up to certain linear transformations. We present a generalization of the result above to the case of generic modulus $m^2$, and show how this is connected to the uniqueness of the representation of sets as a minimal number of arithmetic progression of same difference.