On the set for which 1 is univoque (Q2754967)

For some integer \(k\geq 2\), let \(\varepsilon_k\) be the set of those \(\theta \in \left({1\over {k+1}}, {1\over k}\right)\) for which \(1\) has only one expansion NEWLINE\[NEWLINE1=e_1\theta+e_2\theta^2+\ldots NEWLINE\]NEWLINE with digits \(e_j\in \{0, 1, \ldots , k\},\;(j=1, 2, \ldots)\). In this case \(1\) is said to be an univoque number. The following result is proven: Theorem. For each \(k\in {\mathbb N} \;\;(k\geq 2)\), the set \(\varepsilon_k\) is of Lebesgue measure \(0.\)