Optimal expansions in non-integer bases

For a given positive integer m, let A = {0, 1, . . . , m} and q ∈ (m,m+1). A sequence (ci) = c1c2 . . . consisting of elements in A is called an expansion of x if ∞ i=1 ciq−i = x. It is known that almost every x belonging to the interval [0,m/(q − 1)] has uncountably many expansions. In this paper we study the existence of expansions (di) of x satisfying the inequalities n i=1 diq−i ≥ n i=1 ciq−i , n = 1, 2, . . . , for each expansion (ci) of x.