arXiv subjects

Karma Dajani

Publications and source records attributed to Karma Dajani.

32 records · Page 2Linked to original sources

arXiv2018

Algebraic sums and products of univoque bases

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. Lü, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+λ\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^λ$ contain an interval for all $x\in(0, 1]$ and $λ\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017).

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arXiv2017

Singular non-Pisot Bernoulli convolutions

We identify a family of numbers for which the Bernoulli convolution is singular. Within this family we find two countable collections of Salem numbers in the interval $(1,2)$, and another Salem number and an algebraic integer that is neither Pisot nor Salem in $(1,2)$. It also contains a non-Pisot, non-Salem algebraic number bigger than 3. Hence, we provide the first new explicit examples of singular Bernoulli convolutions since the work of Erdős in 1939.

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arXiv2017

On the points without universal expansions

Let $1<β<2$. Given any $x\in[0, (β-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $β$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_nβ^{-n}.$ 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 $β$-expansion of $x$. Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved that given any $1<β<2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set $V_β$ of points without universal expansions. For any $n\geq 2$, let $β_n$ be the $n$-bonacci number satisfying the following equation: $β^n=β^{n-1}+β^{n-2}+\cdots +β+1.$ Then we have $\dim_{H}(V_{β_n})=1$, 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}.

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arXiv2016

Subshift of finite type and self-similar sets

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of $K$ as well as the set of elements in $K$ with unique codings using the machinery of Mauldin and Williams \cite{MW}. We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of $K$ with multiple codings. Secondly, in the setting of $β$-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik \cite[Theorem 1.8]{MK}. Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera \cite{Barrera}.

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arXiv2016

Shrinking random $β$-transformation

For any $n\geq 3$, let $1<β<2$ be the largest positive real number satisfying the equation $$β^n=β^{n-2}+β^{n-3}+\cdots+β+1.$$ In this paper we define the shrinking random $β$-transformation $K$ and investigate natural invariant measures for $K$, and the induced tranformation of $K$ on a special subset of the domain. We prove that both transformations have a unique measure of maximal entropy. However, the measure induced from the intrinsically ergodic measure for $K$ is not the intrinsically ergodic measure for the induced system.

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arXiv2015

Induced Random $β$-transformation

In this article we study the first return map defined on the switch region induced by the greedy and lazy maps. In particular we study the allowable sequences of return times, and when the first return map is a generalised Lüroth series transformation. We show that there exists a countable collection of disjoint intervals $(\mathcal{I}_{n})_{n=1}^{\infty},$ such that all sequences of return times are permissible if and only if $β\in \mathcal{I}_{n}$ for some $n$. Moreover, we show that there exists a set $M\subseteq(1,2)$ of Hausdorff dimension $1$ and Lebesgue measure zero, for which the first return map is a generalised Lüroth series transformation if and only if $β\in M$.

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arXiv2014

On univoque points for self-similar sets

Let $K\subseteq\mathbb{R}$ be the unique attractor of an iterated function system. We consider the case where $K$ is an interval and study those elements of $K$ with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams \cite{MW}. The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of \cite{DKK}, \cite{K1}, \cite{K2}, and \cite{MK}.

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arXiv2012

Local dimensions for the random beta-transformation

The random beta-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special values of beta and use the projection to obtain results on the local dimension of the Bernoulli convolution.

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arXiv2011

Optimal expansions in non-integer bases

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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arXiv2010

Transformations generating negative $β$-expansions

We introduce a family of dynamical systems that generate negative $β$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of digit sequences that is produced by a typical member of this family of transformations. We discuss the meaning of greedy expansions in the negative sense, and show that there is no transformation in the introduced family of dynamical systems that generates negative greedy. However, if one looks at random algorithms, then it is possible to define a greedy expansion in base $-β$.

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arXiv2008

A natural extension for the greedy beta-transformation with three deleted digits

We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy beta-transformation with three deleted digits. We define a version of the natural extension of the transformation to obtain this expression. We get that the transformation is exact and weakly Bernoulli.

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arXiv2008

Metrical theory for $α$-Rosen fractions

The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the $α$-Rosen fractions. The metrical properties of these $α$-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are closely related to similar region for the 'classical' Rosen fraction. This allows us to unify and generalize results of diophantine approximation from the literature.

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arXiv2008

Two versions of a specific natural extension

We give two versions of the natural extension of a specific greedy beta-transformation with deleted digits. We use the natural extension to obtain an explicit expression for the invariant measure, equivalent to the Lebesgue measure, of this beta-transformation.

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arXiv2006

Entropy quotients and correct digits in number-theoretic expansions

Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the entropies of their dynamical systems. He used detailed knowledge of the continued fraction operator. In 2001, a generalization of Lochs' result was given by Dajani and Fieldsteel in \citeDajF, describing the rate at which the digits of one number-theoretic expansion determine those of another. Their proofs are based on covering arguments and not on the dynamics of specific maps. In this paper we give a dynamical proof for certain classes of transformations, and we describe explicitly the distribution of the number of digits determined when comparing two expansions in integer bases. Finally, using this generalization of Lochs' result, we estimate the unknown entropy of certain number theoretic expansions by comparing the speed of convergence with that of an expansion with known entropy.

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