  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 \[\log k\leq
(\textrm{diam}(A)+1)^{1/3+\epsilon}. \]
