arXiv subjects

Simon Baker

Publications and source records attributed to Simon Baker.

At least 55 records · Page 3Linked to original sources

arXiv2020

On the pair correlations of powers of real numbers

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method.

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arXiv2020

Equidistribution results for sequences of polynomials

Let $(f_n)_{n=1}^{\infty}$ be a sequence of polynomials and $α>1$. In this paper we study the distribution of the sequence $(f_n(α))_{n=1}^{\infty}$ modulo one. We give sufficient conditions for a sequence $(f_n)_{n=1}^{\infty}$ to ensure that for Lebesgue almost every $α>1$ the sequence $(f_n(α))_{n=1}^{\infty}$ has Poissonian pair correlations. In particular, this result implies that for Lebesgue almost every $α>1$, for any $k\geq 2$ the sequence $(α^{n^k})_{n=1}^{\infty}$ has Poissonian pair correlations.

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arXiv2019

Iterated function systems with super-exponentially close cylinders

Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then there are cylinders which are super-exponentially close at all small scales. Several years later, Shmerkin proved an analogous statement for the $L^q$ dimension of self-similar measures in $\mathbb{R}$. With these statements in mind, it is natural to wonder whether there exist iterated function systems that do not contain exact overlaps, yet there are cylinders which are super-exponentially close at all small scales. In this paper we show that such iterated function systems do exist. In fact we prove much more. We prove that for any sequence $(ε_n)_{n=1}^{\infty}$ of positive real numbers, there exists an iterated function system $\{ϕ_i\}_{i\in \mathcal{I}}$ that does not contain exact overlaps and $$\min\left\{|ϕ_{\mathbf{a}}(0)-ϕ_{\mathbf{b}}(0)|: \mathbf{a},\mathbf{b}\in \mathcal{I}^n,\, \mathbf{a}\neq \mathbf{b},\, r_{\mathbf{a}}=r_{\mathbf{b}}\right\}\leq ε_n$$ for all $n\in \mathbb{N}.$

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arXiv2019

Two bifurcation sets arising from the beta transformation with a hole at $0$

Given $β\in(1,2],$ the $β$-transformation $T_β: x\mapsto βx\pmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-valued bifurcation set \[ \mathcal E_β:=\{t\in[0, 1): K_β(t')\ne K_β(t)~\forall t'>t\}, \] where $K_β(t):=\{x\in[0, 1): T_β^n(x)\ge t~\forall n\ge 0\}$ is the survivor set. In this paper we investigate the dimension bifurcation set \[ \mathcal B_β:=\{t\in[0, 1): \dim_H K_β(t')\ne \dim_H K_β(t)~\forall t'>t\}, \] where $\dim_H$ denotes the Hausdorff dimension. We show that if $β\in(1,2]$ is a multinacci number then the two bifurcation sets $\mathcal B_β$ and $\mathcal E_β$ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for $β$ a multinacci number we have $\dim_H(\mathcal E_β\cap[t, 1])=\dim_H K_β(t)$ for any $t\in[0, 1)$. This confirms a conjecture of Kalle et al.~for $β$ a multinacci number.

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arXiv2019

An infinitely generated self-similar set with positive Lebesgue measure and empty interior

Peres and Solomyak asked the question: Do there exist self-similar sets with positive Lebesgue measure and empty interior? This question was answered in the affirmative by Csörnyei et al. They gave a parameterised family of iterated function systems for which almost all of the corresponding self-similar sets satisfied the required properties. They do not however provide an explicit example. Motivated by a desire to construct an explicit example, we in this paper provide an explicit construction of an infinitely generated self-similar set with positive Lebesgue measure and empty interior.

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arXiv2018

On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=λ\mathbf{x} +(1-λ)\mathbf{p}_i,$$ where $λ\in(0,1)$. Given $F$ and $λ$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=\cup_{i=0}^n S_i(X)$. For each $\mathbf{x} \in X$ there exists a sequence $\mathbf{a}\in\{0,\ldots,n\}^{\mathbb{N}}$ satisfying $$\mathbf{x}=\lim_{j\to\infty}(S_{a_1}\circ \cdots \circ S_{a_j})(\mathbf{0}).$$ We call such a sequence a coding of $\mathbf{x}$. In this paper we prove that for any $F$ and $k \in\mathbb{N},$ there exists $δ_k(F)>0$ such that if $λ\in(1-δ_k(F),1),$ then every point in the interior of $X$ has a coding which is $k$-simply normal. Similarly, we prove that there exists $δ_{uni}(F)>0$ such that if $λ\in(1-δ_{uni}(F),1),$ then every point in the interior of $X$ has a coding containing all finite words. For some specific choices of $F$ we obtain lower bounds for $δ_k(F)$ and $δ_{uni}(F)$. We also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction. Our proofs rely on a variation of a well known construction of a normal number due to Champernowne, and an approach introduced by Erdős and Komornik.

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arXiv2018

A General Mass Transference Principle

In this paper we prove a general form of the Mass Transference Principle for $\limsup$ sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in $\mathbb{R}^n$. Our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff $g$-measure statements to Hausdorff $f$-measure statements. This work extends previous results of this type in several distinct directions.

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arXiv2018

Bifurcation sets arising from non-integer base expansions

Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is well-known that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \[ \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. \] Note that $\mathcal B$ is contained in the set $\mathcal{U}^R$ of bases $q\in(1,M+1]$ such that $1\in\mathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash\mathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $\mathcal B\backslash\mathcal{U}^R$ is $\frac{\log 2}{3\log λ^*}\approx 0.368699$, where $λ^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$.

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arXiv2018

Maximising Bernoulli measures and dimension gaps for countable branched systems

Kifer, Peres, and Weiss proved that there exists $c_0>0,$ such that $\dim μ\leq 1-c_0$ for any probability measure $μ$ which makes the digits of the continued fraction expansion i.i.d. random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.

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arXiv2017

Exceptional digit frequencies and expansions in non-integer bases

In this paper we study the set of digit frequencies that are realised by elements of the set of $β$-expansions. The main result of this paper demonstrates that as $β$ approaches $1,$ the set of digit frequencies that occur amongst the set of $β$-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.

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arXiv2017

Entropy, topological transitivity, and dimensional properties of unique $q$-expansions

Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, we study the set $\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\mathbf{U}_q$ of corresponding sequences. It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\in(1, M+1]$ the topological entropy of $\mathcal{U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\mathcal E$ of $q$'s where the function $H$ is not locally constant. Moreover, we show that $\mathcal E$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\mathbf{V}_q, σ),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $\mathcal{U}_q$ coincide for all $q\in(1,M+1]$.

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arXiv2017

Numbers with simply normal $β$-expansions

In [Bak] the first author proved that for any $β\in (1,β_{KL})$ every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion, where $β_{KL}\approx 1.78723$ is the Komornik-Loreti constant. This result is complemented by an observation made in [JSS], where it was shown that whenever $β\in (β_T, 2]$ there exists an $x\in(0,\frac{1}{β-1})$ with a unique $β$-expansion, and this expansion is not simply normal. Here $β_T\approx 1.80194$ is the unique zero in $(1,2]$ of the polynomial $x^3-x^2-2x+1$. This leaves a gap in our understanding within the interval $[β_{KL}, β_T]$. In this paper we fill this gap and prove that for any $β\in (1,β_T],$ every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion. For completion, we provide a proof that for any $β\in(1,2)$, Lebesgue almost every $x$ has a simply normal $β$-expansion. We also give examples of $x$ with multiple $β$-expansions, none of which are simply normal. Our proofs rely on ideas from combinatorics on words and dynamical systems.

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arXiv2017

Root sets of polynomials and power series with finite choices of coefficients

Given $H\subseteq \mathbb{C}$ two natural objects to study are the set of zeros of polynomials with coefficients in $H$, $$\{z\in \mathbb{C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum_{n=0}^{k}a_{n}z^n=0\},$$ and the set of zeros of power series with coefficients in $H$, $$\{z\in\mathbb{C}: \exists (a_n)\in H^{\mathbb{N}}, \sum_{n=0}^{\infty} a_nz^n=0\}.$$ In this paper we consider the case where each element of $H$ has modulus $1$. The main result of this paper states that for any $r\in(1/2,1),$ if $H$ is $2\cos^{-1}(\frac{5-4|r|^2}{4})$-dense in $S^1,$ then the set of zeros of polynomials with coefficients in $H$ is dense in $\{z\in \mathbb{C}: |z|\in [r,r^{-1}]\},$ and the set of zeros of power series with coefficients in $H$ contains the annulus $\{z\in \mathbb{C}: |z|\in[r,1)\}$. These two statements demonstrate quantitatively how the set of polynomial zeros/power series zeros fill out the natural annulus containing them as $H$ becomes progessively more dense.

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arXiv2017

Digit frequencies and self-affine sets with non-empty interior

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $β\in(1,1.787\ldots)$ then every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion. We also prove that if $β\in(1,\frac{1+\sqrt{5}}{2})$ then every $x\in(0,\frac{1}{β-1})$ has a $β$-expansion for which the digit frequency does not exist, and a $β$-expansion with limiting frequency of zeros $p$, where $p$ is any real number sufficiently close to $1/2$. For a class of planar self-affine sets we show that if the horizontal contraction lies in a certain parameter space and the vertical contractions are sufficiently close to $1,$ then every nontrivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and give rise to new examples of self-affine sets with non-empty interior. One particular strength of our approach is that it allows for different rates of contraction in the vertical direction.

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arXiv2016

On the regularity of the generalised golden ratio function

Given a finite set of real numbers $A$, the generalised golden ratio is the unique real number $\mathcal{G}(A) > 1$ for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases. We show that $\mathcal{G}(A)$ varies continuously with the alphabet $A$ (of fixed size). What is more, we demonstrate that as we vary a single parameter $m$ within~$A$, the generalised golden ratio function may behave like $m^{1/h}$ for any positive integer $h$. These results follow from a detailed study of $\mathcal{G}(A)$ for ternary alphabets, building upon the work of Komornik, Lai, and Pedicini (2011). We give a new proof of their main result, that is we explicitly calculate the function $\mathcal{G}(\{0,1,m\})$. (For a ternary alphabet, it may be assumed without loss of generality that $A = \{0,1,m\}$ with $m\in(1,2)]$.) We also study the set of $m \in (1,2]$ for which $\mathcal{G}(\{0,1,m\})=1+\sqrt{m},$ we prove that this set is uncountable and has Hausdorff dimension~$0$. We show that the function mapping $m$ to $\mathcal{G}(\{0,1,m\})$ is of bounded variation yet has unbounded derivative. Finally, we show that it is possible to have unique expansions as well as points with precisely two expansions at the generalised golden ratio.

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arXiv2016

On periodic representations in non-Pisot bases

We study periodic expansions in positional number systems with a base $β\in\C,\ |β|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(β),$ such that all elements of $\Q(β)$ admit at least one eventually periodic representation with digits in $\A$. In this paper we prove a general result that guarantees the existence of such an $\A$. This result implies the existence of such an $\A$ when $β$ is a rational number or an algebraic integer with no conjugates of modulus $1$. We also consider eventually periodic representations of elements of $\Q(β)$ for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of $\Q(β)$ admits such a representation then $β$ must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt \cite{Schmidt}.

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arXiv2016

Unique expansions and intersections of Cantor sets

To each $α\in(1/3,1/2)$ we associate the Cantor set $$Γ_α:=\Big\{\sum_{i=1}^{\infty}ε_{i}α^i: ε_i\in\{0,1\},\,i\geq 1\Big\}.$$ In this paper we consider the intersection $Γ_α\cap (Γ_α+ t)$ for any translation $t\in\mathbb{R}$. We pay special attention to those $t$ with a unique $\{-1,0,1\}$ $α$-expansion, and study the set $$D_α:=\{\dim_H(Γ_α\cap (Γ_α+ t)):t \textrm{ has a unique }\{-1,0,1\}\,α\textrm{-expansion}\}.$$ We prove that there exists a transcendental number $α_{KL}\approx 0.39433\ldots$ such that: $D_α$ is finite for $α\in(α_{KL},1/2),$ $D_{α_{KL}}$ is infinitely countable, and $D_α$ contains an interval for $α\in(1/3,α_{KL}).$ We also prove that $D_α$ equals $[0,\frac{\log 2}{-\log α}]$ if and only if $α\in (1/3,\frac {3-\sqrt{5}}{2}].$ As a consequence of our investigation we prove some results on the possible values of $\dim_{H}(Γ_α\cap (Γ_α+ t))$ when $Γ_α\cap (Γ_α+ t)$ is a self-similar set. We also give examples of $t$ with a continuum of $\{-1,0,1\}$ $α$-expansions for which we can explicitly calculate $\dim_{H}(Γ_α\cap(Γ_α+t)),$ and for which $Γ_α\cap (Γ_α+t)$ is a self-similar set. We also construct $α$ and $t$ for which $Γ_α\cap (Γ_α+ t)$ contains only transcendental numbers. Our approach makes use of digit frequency arguments and a lexicographic characterisation of those $t$ with a unique $\{-1,0,1\}$ $α$-expansion.

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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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