On β-expansions of unity for rational and transcendental numbers β

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Abstract We investigate the sequence of integers x 1, x 2, x 3, … lying in {0, 1, …, [β]} in a so-called Rényi β-expansion of unity 1 = $\sum\limits_{j = 1}^\infty {x_j \beta ^{ - j} } $ for rational and transcendental numbers β > 1. In particular, we obtain an upper bound for two strings of consecutive zeros in the β-expansion of unity for rational β. For transcendental numbers β which are badly approximable by algebraic numbers of every large degree and bounded height, we obtain an upper bound for the Diophantine exponent of the sequence X = (xj)j=1∞ in terms of β.