arXiv subjects

Simon Baker

Publications and source records attributed to Simon Baker.

69 records · Page 4Linked to original sources

arXiv2015

Inhomogeneous self-similar sets with overlaps

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this `expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $SO(d)$ for $d\geq 3$. We also obtain new upper bounds for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the \emph{weak separation property} is satisfied, ie. the overlaps are controllable, then the `expected formula' does hold.

math.CA↗

arXiv2015

Approximation properties of $β$-expansions II

Given $β\in(1,2)$ and $x\in[0,\frac{1}{β-1}]$, a sequence $(ε_{i})_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is called a $β$-expansion for $x$ if $$x=\sum_{i=1}^{\infty}\frac{ε_{i}}{β^{i}}.$$ In a recent article the author studied the quality of approximation provided by the finite sums $\sum_{i=1}^{n}ε_{i}β^{-i}$ \cite{Bak}. In particular, given $β\in(1,2)$ and $Ψ:\mathbb{N}\to\mathbb{R}_{\geq 0},$ we associate the set $$W_β(Ψ):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}\bigcup_{(ε_{i})_{i=1}^{n}\in\{0,1\}^{n}}\Big[\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}},\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}+Ψ(n)\Big].$$ Alternatively, $W_β(Ψ)$ is the set of $x\in \mathbb{R}$ such that for infinitely many $n\in\mathbb{N},$ there exists a sequence $(ε_{i})_{i=1}^{n}$ satisfying the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}\leq Ψ(n).$$ If $\sum_{n=1}^{\infty}2^{n}Ψ(n)<\infty$ then $W_β(Ψ)$ has zero Lebesgue measure. We call a $β\in(1,2)$ approximation regular, if $\sum_{n=1}^{\infty}2^{n}Ψ(n)=\infty$ implies $W_β(Ψ)$ is of full Lebesgue measure within $[0,\frac{1}{β-1}]$. The author conjectured in \cite{Bak} that almost every $β\in(1,2)$ is approximation regular. In this paper we make a significant step towards proving this conjecture. The main result of this paper is the following statement: given a sequence of positive real numbers $(ω_{n})_{n=1}^{\infty},$ which satisfy $\lim_{n\to\infty} ω_{n}=\infty$, then for Lebesgue almost every $β\in(1.497\ldots,2)$ the set $W_β(ω_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,\frac{1}{β-1}]$. Here the sequence $(ω_{n})_{n=1}^{\infty}$ should be interpreted as a sequence tending to infinity at a very slow rate.

math.NT↗

arXiv2015

On small bases for which $1$ has countably many expansions

Let $q\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $(δ_i)_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ satisfying $$ x=\sum_{i=1}^\infty\frac{δ_i}{q^i}.$$ Let $\mathcal{B}_{\aleph_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let $\mathcal{B}_{1, \aleph_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \cite{Sidorov6} it was shown that $\min\mathcal{B}_{\aleph_0}=\min\mathcal{B}_{1,\aleph_0}=\frac{1+\sqrt{5}}{2},$ and in \cite{Baker} it was shown that $\mathcal{B}_{\aleph_0}\cap(\frac{1+\sqrt{5}}{2}, q_1]=\{ q_1\}$, where $q_1(\approx1.64541)$ is the positive root of $x^6-x^4-x^3-2x^2-x-1=0$. In this paper we show that the second smallest point of $\mathcal{B}_{1,\aleph_0}$ is $q_3(\approx1.68042)$, the positive root of $x^5-x^4-x^3-x+1=0$. Enroute to proving this result we show that $\mathcal{B}_{\aleph_0}\cap(q_1, q_3]=\{ q_2, q_3\}$, where $q_2(\approx1.65462)$ is the positive root of $x^6-2x^4-x^3-1=0$.

math.NT↗

arXiv2014

Dynamical properties of S-gap shifts and other shift spaces

We study the dynamical properties of certain shift spaces. To help study these properties we introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified shift is both BSM and balanced, and any balanced shift is BSM. However, as we will demonstrate, there are examples of shifts which are BSM but not balanced. We also study the measure theoretic properties of balanced shifts. We show that a shift space admits a Gibbs state if and only if it is balanced. Restricting ourselves to $S$-gap shifts, we relate certain dynamical properties of an $S$-gap shift to combinatorial properties from expansions in non-integer bases. This identification allows us to use the machinery from expansions in non-integer bases to give straightforward constructions of $S$-gap shifts with certain desirable properties. We show that for any $q\in(0,1)$ there is an $S$-gap shift which has the specification property and entropy $q$. We also use this identification to address the question, for a given $q\in(0,1),$ how many $S$-gap shifts exist with entropy $q?$ For certain exceptional values of $q$ there is a unique $S$-gap shift with this entropy.

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arXiv2014

On the distribution of powers of real numbers modulo 1

Given a strictly increasing sequence of positive real numbers tending to infinity $(q_{n})_{n=1}^{\infty}$, and an arbitrary sequence of real numbers $(r_{n})_{n=1}^{\infty}.$ We study the set of $α\in(1,\infty)$ for which $\lim_{n\to\infty}\|α^{q_{n}}-r_{n}\|= 0$. In \cite{Dub} Dubickas showed that whenever $\lim_{n\to\infty}(q_{n+1}-q_{n})=\infty,$ there always exists a transcendental $α$ for which $\lim_{n\to\infty}\|α^{q_{n}}-r_{n}\|= 0.$ Adapting the approach of Bugeaud and Moshchevitin \cite{BugMos}, we improve upon this result and show that whenever $\lim_{n\to\infty}(q_{n+1}-q_{n})=\infty,$ the set of $α\in(1,\infty)$ satisfying $\lim_{n\to\infty}\|α^{q_{n}}-r_{n}\|= 0$ is a dense set of Hausdorff dimension $1$.

math.NT↗

arXiv2014

Approximation properties of $β$-expansions

Let $β\in(1,2)$ and $x\in [0,\frac{1}{β-1}]$. We call a sequence $(ε_{i})_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ a $β$-expansion for $x$ if $x=\sum_{i=1}^{\infty}ε_{i}β^{-i}$. We call a finite sequence $(ε_{i})_{i=1}^{n}\in\{0,1\}^{n}$ an $n$-prefix for $x$ if it can be extended to form a $β$-expansion of $x$. In this paper we study how good an approximation is provided by the set of $n$-prefixes. Given $Ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$, we introduce the following subset of $\mathbb{R}$, $$W_β(Ψ):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^\infty\bigcup_{(ε_{i})_{i=1}^{n}\in\{0,1\}^{n}}\Big[\sum_{i=1}^{n}\frac{ε_i}{β^{i}}, \sum_{i=1}^n\frac{ε_i} {β^i}+Ψ(n)\Big]$$ In other words, $W_β(Ψ)$ is the set of $x\in\mathbb{R}$ for which there exists infinitely many solutions to the inequalities $$0\leq x-\sum_{i=1}^{n}\frac{ε_{i}}{β^{i}}\leq Ψ(n).$$ When $\sum_{n=1}^{\infty}2^{n}Ψ(n)<\infty$ the Borel-Cantelli lemma tells us that the Lebesgue measure of $W_β(Ψ)$ is zero. When $\sum_{n=1}^{\infty}2^{n}Ψ(n)=\infty,$ determining the Lebesgue measure of $W_β(Ψ)$ is less straightforward. Our main result is that whenever $β$ is a Garsia number and $\sum_{n=1}^{\infty}2^{n}Ψ(n)=\infty$ then $W_β(Ψ)$ is a set of full measure within $[0,\frac{1}{β-1}]$. Our approach makes no assumptions on the monotonicity of $Ψ,$ unlike in classical Diophantine approximation where it is often necessary to assume $Ψ$ is decreasing.

math.NT↗

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}.

math.DS↗

arXiv2014

Badly approximable numbers for sequences of balls

It is a classical result from Diophantine approximation that the set of badly approximable numbers has Lebesgue measure zero. In this paper we generalise this result to more general sequences of balls. Given a countable set of closed $d$-dimensional Euclidean balls $\{B(x_{i},r_{i})\}_{i=1}^{\infty},$ we say that $α\in \mathbb{R}^{d}$ is a badly approximable number with respect to $\{B(x_{i},r_{i})\}_{i=1}^{\infty}$ if there exists $κ(α)>0$ and $N(α)\in\mathbb{N}$ such that $α\notin B(x_{i},κ(α)r_{i})$ for all $i\geq N(α)$. Under natural conditions on the set of balls, we prove that the set of badly approximable numbers with respect to $\{B(x_{i},r_{i})\}_{i=1}^{\infty}$ has Lebesgue measure zero. Moreover, our approach yields a new proof that the set of badly approximable numbers has Lebesgue measure zero.

math.NT↗

arXiv2014

On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Let $λ_{1},\ldots,λ_{n}$ be real numbers in $(0,1)$ and $p_{1},\ldots,p_{n}$ be points in $\mathbb{R}^{d}$. Consider the collection of maps $f_{j}:\mathbb{R}^{d}\to\mathbb{R}^{d} $ given by $$f_{j}(x)=λ_{j} x +(1-λ_{j})p_{j}.$$ It is a well known result that there exists a unique compact set $Λ\subset \mathbb{R}^{d}$ satisfying $Λ=\cup_{j=1}^{n} f_{j}(Λ).$ Each $x\in Λ$ has at least one coding, that is a sequence $(ε_{i})_{i=1}^{\infty}\in \{1,\ldots,n\}^{\mathbb{N}}$ that satisfies $\lim_{N\to\infty}f_{ε_{1}}\cdots f_{ε_{N}} (0)=x.$ We study the size and complexity of the set of codings of a generic $x\in Λ$ when $Λ$ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every $x\inΛ$ has a continuum of codings. We also show that almost every $x\inΛ$ has a universal coding. Our work makes no assumptions on the existence of holes in $Λ$ and improves upon existing results when it is assumed $Λ$ contains no holes.

math.DS↗

arXiv2014

Expansions in non-integer bases: lower order revisited

Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\varepsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if \[ x=\sum_{i=1}^{\infty}\varepsilon_iq^{-i}. \] For any $k\in\mathbb N$, let $\mathcal B_k$ denote the set of $q$ such that there exists $x$ with exactly $k$ expansions in base $q$. In [12] it was shown that $\min\mathcal B_2=q_2\approx 1.71064$, the appropriate root of $x^{4}=2x^{2}+x+1$. In this paper we show that for any $k\geq 3$, $\min\mathcal B_k=q_f\approx1.75488$, the appropriate root of $x^3=2x^2-x+1$.

math.NT↗

arXiv2013

On universal and periodic $β$-expansions, and the Hausdorff dimension of the set of all expansions

In this paper we study the topology of a set naturally arising from the study of $β$-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of $β$-expansions are equal and explicitly calculable.

math.DS↗

arXiv2013

On small bases which admit countably many expansions

Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(ε_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if x=\sum_{i=1}^{\infty}ε_iq^{-i}. Let $\mathcal{B}_{\aleph_{0}}$ denote the set of $q$ for which there exists $x$ with exactly $\aleph_{0}$ expansions in base $q$. In \cite{EHJ} it was shown that $\min\mathcal{B}_{\aleph_{0}}=\frac{1+\sqrt{5}}{2}.$ In this paper we show that the smallest element of $\mathcal{B}_{\aleph_{0}}$ strictly greater than $\frac{1+\sqrt{5}}{2}$ is $q_{\aleph_{0}}\approx1.64541$, the appropriate root of $x^6=x^4+x^3+2x^2+x+1$. This leads to a full dichotomy for the number of possible $q$-expansions for $q\in (\frac{1+\sqrt{5}}{2},q_{\aleph_{0}})$. We also prove some general results regarding $\mathcal{B}_{\aleph_{0}}\cap[\frac{1+\sqrt{5}}{2},q_{f}],$ where $q_{f}\approx 1.75488$ is the appropriate root of $x^{3}=2x^{2}-x+1.$ Moreover, the techniques developed in this paper imply that if $x\in [0,\frac{1}{q-1}]$ has uncountably many $q$-expansions then the set of $q$-expansions for $x$ has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case.

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arXiv2012

Generalised golden ratios over integer alphabets

It is a well known result that for $β\in(1,\frac{1+\sqrt{5}}{2})$ and $x\in(0,\frac{1}{β-1})$ there exists uncountably many $(ε_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ such that $x=\sum_{i=1}^{\infty}ε_{i}β^{-i}.$ When $β\in(\frac{1+\sqrt{5}}{2},2]$ there exists $x\in (0,\frac{1}{β-1})$ for which there exists a unique $(ε_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ such that $x=\sum_{i=1}^{\infty}ε_{i}β^{-i}.$ In this paper we consider the more general case when our sequences are elements of ${0,...,m}^{\mathbb{N}}.$ We show that an analogue of the golden ratio exists and give an explicit formula for it.

math.DS↗

arXiv2012

The growth rate and dimension theory of beta-expansions

In a recent paper of Feng and Sidorov they show that for $β\in(1,\frac{1+\sqrt{5}}{2})$ the set of $β$-expansions grows exponentially for every $x\in(0,\frac{1}{β-1})$. In this paper we study this growth rate further. We also consider the set of $β$-expansions from a dimension theory perspective.

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arXiv2012

A multifractal zeta function for cookie cutter sets

Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectrum for a class of measures.

math.DS↗