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Register a new userBeta Expansions for Regular Pisot Numbers
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Maysum Panju
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A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is known that if the base is a Pisot number, then this expansion will always be finite or periodic. Some work has been done to learn more about these expansions, but in general these expansions were not explicitly known. In this paper, we present a complete list of the greedy beta expansions of 1 where the base is any regular Pisot number less than 2, revealing a variety of remarkable patterns. We also answer a conjecture of Boyds regarding cyclotomic co-factors for greedy expansions.
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It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i in [0,q)$ satisfies the equality $sum_{i=1}^infty c_iq^{-i}=1$. The set of such univoque numbers has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed $q>1$ the set $mathcal{U}_q$ of real numbers $x$ having a unique representation of the form $sum_{i=1}^infty c_iq^{-i}=x$ with integers $c_i$ belonging to $[0,q)$. We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases $q$ for which $mathcal{U}_q$ is closed or even a Cantor set. We also study the set $mathcal{U}_q$ consisting of all sequences $(c_i)$ of integers $c_i in [0,q)$ such that $sum_{i=1}^{infty} c_i q^{-i} in mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q mapsto mathcal{U}_q$ (defined on $(1, infty)$) is constant in a neighborhood of $r$ and the numbers $q >1$ for which $mathcal{U}_q$ is a subshift or a subshift of finite type.
Let $M$ be a positive integer and $qin (1, M+1]$. A $q$-expansion of a real number $x$ is a sequence $(c_i)=c_1c_2cdots$ with $c_iin {0,1,ldots, M}$ such that $x=sum_{i=1}^{infty}c_iq^{-i}$. In this paper we study the set $mathcal{U}_q^j$ consisting of those real numbers having exactly $j$ $q$-expansions. Our main result is that for Lebesgue almost every $qin (q_{KL}, M+1), $ we have $$dim_{H}mathcal{U}_{q}^{j}leq max{0, 2dim_Hmathcal{U}_q-1}text{ for all } jin{2,3,ldots}.$$ Here $q_{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $jin{2,3,ldots},$ the function mapping $q$ to $dim_{H}mathcal{U}_{q}^{j}$ is not continuous.
A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base $3$ or $4$ expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Grahams problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in $[0, 1]$ who do not contain some digit in their $b$-expansion for all $b geq 3$ has zero Hausdorff dimension.
We investigate the complex spectra [ X^{mathcal A}(beta)=left{sum_{j=0}^na_jbeta^j : nin{mathbb N}, a_jin{mathcal A}right} ] where $beta$ is a quadratic or cubic Pisot-cyclotomic number and the alphabet $mathcal A$ is given by $0$ along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut and project method.
For $beta > 1$, a sequence $(c_n)_{n geq 1} in mathbb{Z}^{mathbb{N}^+}$ with $0 leq c_n < beta$ is the emph{beta expansion} of $x$ with respect to $beta$ if $x = sum_{n = 1}^infty c_nbeta^{-n}$. Defining $d_beta(x)$ to be the greedy beta expansion of $x$ with respect to $beta$, it is known that $d_beta(1)$ is eventually periodic as long as $beta$ is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with $text{trace}(beta) leq 15$. We give examples of numbers with $text{trace}(beta) leq 15$ whose expansions have period and preperiod lengths exceeding $10^{10}$, yet are still eventually periodic.
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