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

Year, pages: 2011, 705 - 716

We investigate the sequence of integers $x₁, x₂, x₃, …$ lying in $\{0,1,…,[β]\}$ in a so-called Rényi $\beta$-expansion of unity $1=\sum_{j=1}^{\infty} x_j \beta^{-j}$ for rational and transcendental numbers $\beta>1$. In particular, we obtain an upper bound for two strings of consecutive zeros in the $\beta$-expansion of unity for rational $\beta$. For transcendental numbers $\beta$ 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 $X=(x_j)_{j=1}^{\infty}$ in terms of $\beta$.

Dubickas, A. 2011. On $β$-expansions of unity for rational and transcendental numbers $β$. In Mathematica Slovaca, vol. 61, no.5, pp. 705-716. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0040-3

Dubickas, A. (2011). On $β$-expansions of unity for rational and transcendental numbers $β$. Mathematica Slovaca, 61(5), 705-716. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0040-3