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Abstract: Let $\beta>1$ and let $m>\be$ be an integer. Each $x\inI \be:=0,\frac{m-1}{\beta-1}$ can be represented in the form \x=\sum {k=1}^\infty \epsilon k\beta^{-k}, \ where$\epsilon k\in\{0,1,

.,m-1\}$ for all $k$ a $\beta$-expansion of $x$. It isknown that a.e. $x\in I \beta$ has a continuum of distinct $\beta$-expansions.In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$this continuum has one and the same growth rate. We also link this rate to theLebesgue-generic local dimension for the Bernoulli convolution parametrized by$\beta$.When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansionsgrows exponentially for every internal $x$.



Author: De-Jun Feng, Nikita Sidorov

Source: https://arxiv.org/







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