# On the points without universal expansions

## TL;DR

This paper investigates the set of points lacking universal beta-expansions, showing that for certain algebraic numbers like the n-bonacci numbers, this set has full Hausdorff dimension, and explores related dynamical systems.

## Contribution

It establishes that the set of points without universal beta-expansions has Hausdorff dimension one for specific algebraic bases, extending previous results to new classes of numbers.

## Key findings

- Hausdorff dimension of V_beta is 1 for beta equal to n-bonacci numbers
- Results extend to some other algebraic numbers
- Applications to the Hausdorff dimension of survivor sets in dynamical systems

## Abstract

Let $1<\beta<2$. Given any $x\in[0, (\beta-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $\beta$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_n\beta^{-n}.$ For any $k\geq 1$ and any $(b_1b_2\cdots b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1b_2\cdots b_k$, then we call $(a_n)$ a universal $\beta$-expansion of $x$. Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved that given any $1<\beta<2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set $V_{\beta}$ of points without universal expansions. For any $n\geq 2$, let $\beta_n$ be the $n$-bonacci number satisfying the following equation: $\beta^n=\beta^{n-1}+\beta^{n-2}+\cdots +\beta+1.$ Then we have $\dim_{H}(V_{\beta_n})=1$, where $\dim_{H}$ denotes the Hausdorff dimension. Similar results are still available for some other algebraic numbers. As a corollary, we give some results of the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper \cite{KarmaKan}.