Optimal expansions in non-integer bases

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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∞ cᵢ q⁻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 ∑ᵢ₌₁ⁿ dᵢq⁻ⁱ ≥ ∑ᵢ₌₁ⁿ cᵢ q⁻ⁱ, n=1,2,... for each expansion (cᵢ) of x.