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Universal $β$-expansions

Nikita Sidorov
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Abstract

Given $β\in(1,2)$, a $β$-expansion of a real $x$ is a power series in base $β$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of $β$-expansions. Our main result is that for any $β\in(1,2)$ and a.e. $x\in (0,1)$ there always exists a universal $β$-expansion of $x$ in the sense of Erdös and Komornik, i.e., a $β$-expansion whose complexity function is $2^n$. We also study some questions related to the points having less than a full branching continuum of $β$-expansions and also normal $β$-expansions.

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Nikita Sidorov. 2002-09-19. Universal $β$-expansions. https://arxiv.org/abs/math/0209247

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