Periodicity of complementing multisets

TL;DR

This paper proves that for certain multisets of integers, the complementing multiset must be periodic and provides an upper bound on its smallest period based on the multiset's diameter and weight sum.

Contribution

It establishes a Biro-type upper bound on the period of complementing multisets, extending previous results with explicit bounds depending on multiset parameters.

Findings

Complementing multisets are necessarily periodic.

The smallest period's logarithm is bounded by a function of the multiset's diameter.

The bound depends on the multiset's diameter and weight sum, with explicit constants.

Abstract

Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$ and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We obtain the Biro-type upper bound for the smallest such period of $B$: Let $\epsilon>0$. We assume that $\textrm{diam}(A)\ge n_0(\epsilon)$ and that $\sum_{a\in A}w_A(a)\leq (\textrm{diam}(A)+1)^{c}$, where $c$ is any constant such that $c< 100\log2-2$. Then $B$ is periodic with period \[\log k\leq (\textrm{diam}(A)+1)^{1/3+\epsilon}. \]