Related papers: On digit frequencies in {\beta}-expansions
For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…
The spectrum of a real number $\beta>1$ is the set $X^{m}(\beta)$ of $p(\beta)$ where $p$ ranges over all polynomials with coefficients restricted to ${\mathcal A}=\{0,1,\dots,m\}$. For a quadratic Pisot unit $\beta$, we determine the…
We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log…
In this paper, we determine depth spectra of all repeated-root $(\alpha+\gamma \beta)$-constacyclic codes of arbitrary lengths over a finite commutative chain ring $\mathcal{R},$ where $\alpha$ is a non-zero element of the Teichm\"{u}ller…
We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and…
This technical note is on digital filters for the high-fidelity estimation of a sinusoidal signal's frequency in the presence of additive noise. The complex noise is assumed to be white (i.e. uncorrelated) however it need not be Gaussian.…
We present a study of the problem of finiteness of the $\beta$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $\beta$-expansion of 1 is not a non-decreasing sequence.…
Let $\beta>1$. For $x \in [0,\infty)$, we have so-called the $\beta$-expansion of $x$ in base $\beta$ as follows: $$x= \sum_{j \leq k} x_{j}\beta^{j} = x_{k}\beta^{k}+ \cdots + x_{1}\beta+x_{0}+x_{-1}\beta^{-1} + x_{-2}\beta^{-2} + \cdots$$…
In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let $T_{\beta}$ be the beta-transformation with base $\beta>1$, $\{x_{n}\}_{n\geq 1}$ be a…
The beta transformation is the iterated map $\beta x\,\mod1$; it generates the base-$\beta$ expansion of a real number x. Every iterated piece-wise monotonic map is topologically conjugate to the beta transformation. For all but a countable…
Zaremba's conjecture concerns a formation of continued fraction expansions for rational numbers with partial quotient bounded by an absolute constant. We present asymptotic estimates for the size of $\epsilon$-thickening of certain fractal…
We generalize the generalized-squeezing problem to include fractional values of the squeezing order $n$. This approach allows us to determine the locations of critical points at which qualitative changes in behaviour occur and accurately…
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one…
We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation,…
Matrix convexity generalizes convexity to the dimension free setting and has connections to many mathematical and applied pursuits including operator theory, quantum information, noncommutative optimization, and linear control systems. In…
The (-\beta)-integers are natural generalisations of the \beta-integers, and thus of the integers, for negative real bases. They can be described by infinite words which are fixed points of anti-morphisms. We show that they are not…
We study the growth of fractal clusters in the Dielectric Breakdown Model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent…
It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta…
Power spectral density scaling with frequency $f$ as $1/f^\beta$ and $\beta \approx 1$ is widely found in natural and socio-economic systems. Consequently, it has been suggested that such self-similar spectra reflect the universal dynamics…
This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $\beta$-expansions. More precisely, let $([0,1),T_{\beta})$ be the $\beta$-dynamical system for a general $\beta>1$ and…