Periodicity of complementing multisets

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 $\text{diam}(A)\ge n_0(\epsilon )$ and that ${\sum }_{a\in A}{w}_A(a)\le (\text{diam}(A)+1{)}^c$, where $c$ is any constant such that $c<100\log 2-2$. Then $B$ is periodic with period \[\log k\leq (\textrm{diam}(A)+1)^{1/3+ε}. \]

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