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Let $\\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\\cdots,\\aleph_0$ let $\\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\\in \\mathcal{B}_k$ let $\\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. 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