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arXiv · 1011.5220

Optimal expansions in non-integer bases

Karma Dajani·Martijn de Vries·Vilmos Komornik·Paola Loreti
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Abstract

For a given positive integer $m$, let $A=\set{0,1,...,m}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 ...$ consisting of elements in $A$ is called an expansion of $x$ if $\sum_{i=1}^{\infty} c_i q^{-i}=x$. It is known that almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many expansions. In this paper we study the existence of expansions $(d_i)$ of $x$ satisfying the inequalities $\sum_{i=1}^n d_iq^{-i} \geq \sum_{i=1}^n c_i q^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$.

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BibTeXRIS

Karma Dajani, Martijn de Vries, Vilmos Komornik, Paola Loreti. 2011-05-15. Optimal expansions in non-integer bases. https://arxiv.org/abs/1011.5220

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