Finite beta-expansions of natural numbers

Let beta > 1. For x is an element of [0, infinity), we have so-called a beta-expansion of x in base beta as follows: x = Sigma(j <= k) x(j)beta(j) = x(k)beta(k) + center dot center dot center dot + x(1)beta + x(0) + x(-1)beta( -1) + x(-2)beta(-2) + center dot center dot center dot where k is an element of Z, beta(k) <= x < beta(k+1), x(j) is an element of Z boolean AND [0, beta) for all j < k and Sigma(j <= n) x(j)beta(j) < beta(n+1) for all n <= k. In this paper, we give a sufficient condition (for beta) such that each element of N has a finite beta -expansion in base beta. Moreover we also find a beta with this finiteness property which does not have positive finiteness property.