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 $\varepsilon>0$. We assume that $diam(A)\ge n_0(\varepsilon)$ and that $\sum_{a\in A}w_A(a)\leq (diam(A)+1)^{c}$, where $c$ is any constant such that $c< 100\log2-2$. Then $B$ is periodic with period $\log k\leq (diam(A)+1)^{\frac{1}{3}+\varepsilon}$.