Periodicity of complementing multisets

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 $ε>0$. We assume that $\textrm{diam}(A)\ge n_0(ε)$ 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 $\log k\leq (\textrm{diam}(A)+1)^{1/3+ε}.$

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