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In this paper we investigate the local dimension of U (x) and prove a 'variation principle' for unique non-integer base expansions. We also determine the critical values of U (x) such that when x passes the first critical value the set U (x) changes from a set with positive Hausdorff dimension to a countable set, and when x passes the second critical value the set U (x) changes from an infinite set to a singleton. Denote by U (x) the set of all unique q-expansions of x for q is an element of U (x). We give the Hausdorff dimension of U (x) and show that the dimensional function x Bar right arrow dim(H) U (x) is a non-increasing Devil's staircase. Finally, we investigate the topological structure of U (x). Although the set U (1) has no isolated points, we prove that for typical x > 0 the set U (x) contains isolated points. (C) 2020 Elsevier Inc. 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