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papers

Publications (69)

math.DS2022

Recurrence rates for shifts of finite type

Demi Allen, Simon Baker, BalÃ¡zs BÃ¡rÃ¡ny

Let $I^£_{A}$ be a topologically mixing shift of finite type, let $I ¨ : I^£_{A} \to I^£_{A}$ be the usual left-shift, and let $I^¼$ be the Gibbs measure for a HÃ¶lder continuous potential…

math.DS2025

Distinct dimensions for attractors of bi-Lipschitz iterated function systems

Simon Baker, Amlan Banaji, De-Jun Feng +2

In this paper, we construct an iterated function system on the line consisting of two bi-Lipschitz contractions whose attractor has distinct lower, Hausdorff, lower box, upper box,…

math.NT2016

On periodic representations in non-Pisot bases

Simon Baker, Zuzana MasÃ¡kovÃ¡, Edita PelantovÃ¡ +1

We study periodic expansions in positional number systems with a base $Î²\in\C,\ |Î²|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determ…

math.DS2018

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

Simon Baker, Derong Kong

Let $F = {p_{0}, \dots, p_{n}}$ be a collection of points in $R^{d} .$ The set $F$ naturally gives rise to a family of iterated function systems consisting of co…

math.DS2017

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

Rafael Alcaraz Barrera, Simon Baker, Derong Kong

Let $M$ be a positive integer and $q \in (1, M + 1] .$ We consider expansions of real numbers in base $q$ over the alphabet ${0, \dots, M}$ . In particular, we study the set $\mathcal{…

math.NT2016

On the regularity of the generalised golden ratio function

Simon Baker, Wolfgang Steiner

Given a finite set of real numbers $A$ , the generalised golden ratio is the unique real number $G (A) > 1$ for which we only have trivial unique expansions in smaller base…

math.NT2021

Metric results for numbers with multiple $q$ -expansions

Simon Baker, Yuru Zou

Let $M$ be a positive integer and $q \in (1, M + 1]$ . A $q$ -expansion of a real number $x$ is a sequence $(c_{i}) = c_{1} c_{2} \dots$ with $c_{i} \in {0, 1, \dots, M}$ such that $x=\sum_{i=1}^{…

math.DS2021

On normal numbers and self-similar measures

Amir Algom, Simon Baker, Pablo Shmerkin

Let ${f_{i} (x) = s_{i} \cdot x + t_{i}}$ be a self-similar IFS on $R$ and let $I^^{2} > 1$ be a Pisot number. We prove that if $\frac{l o g ∣ s _{i} ∣}{l o g I ^ ^{2}} \in / Q$ …

math.DS2025

On finitely many base $q$ expansions

Simon Baker, George Bender

Given some integer $m \geq 3$ , we find the first explicit collection of countably many intervals in $(1, 2)$ such that for any $q$ in one of these intervals, the set of points with…

math.DS2023

Spectral gaps and Fourier dimension for self-conformal sets with overlaps

Simon Baker, Tuomas Sahlsten

We prove a uniform spectral gap for complex transfer operators near the critical line associated to overlapping $C^{2}$ iterated function systems on the real line satisfying a Unifor…

math.DS2021

New dimension bounds for $I^\pm I^^{2}$ sets

Simon Baker

In this paper we obtain new lower bounds for the upper box dimension of $I^\pm I^^{2}$ sets. As a corollary of our main result, we show that if $I^\pm$ is not a Liouville number and $I^^{2}$ is a…

cs.CL2020

Stylistic Dialogue Generation via Information-Guided Reinforcement Learning Strategy

Yixuan Su, Deng Cai, Yan Wang +4

Stylistic response generation is crucial for building an engaging dialogue system for industrial use. While it has attracted much research interest, existing methods often generate…

math.DS2023

On the cardinality and dimension of the slices of Okamoto's functions

Simon Baker, George Bender

The graphs of Okamoto's functions, denoted by $K_{q}$ , are self-affine fractal curves contained in $[0, 1]^{2}$ , parameterised by $q \in (1, 2)$ . In this paper we consider the cardinalit…

math.DS2019

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

Simon Baker, Nikita Sidorov

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 CsÃ¶rnye…

eess.IV2026

Rethinking Clinical Relevance in Chest X-ray Machine Learning: How Evaluation References Define Performance

Panagiotis Fytas, Ian Selby, Clemens Karner +14

Chest X-ray (CXR) machine learning relies heavily on automated evaluation using reference standards that aim to approximate clinical judgment. However, commonly used report-derived…

math.DS2020

An analogue of Khintchine's theorem for self-conformal sets

Simon Baker

Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurrin…

math.NT2020

On the pair correlations of powers of real numbers

Christoph Aistleitner, Simon Baker

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 Ko…

math.NT2017

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

Simon Baker, Han Yu

Given $H \subseteq C$ 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…

cs.CL2020

Prototype-to-Style: Dialogue Generation with Style-Aware Editing on Retrieval Memory

Yixuan Su, Yan Wang, Simon Baker +4

The ability of a dialog system to express prespecified language style during conversations has a direct, positive impact on its usability and on user satisfaction. We introduce a n…

math.NT2021

Intrinsic Diophantine Approximation for overlapping iterated function systems

Simon Baker

In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then appl…

math.NT2014

Approximation properties of $I^^{2}$ -expansions

Simon Baker

Let $I^^{2} \in (1, 2)$ and $x \in [0, \frac{1}{I ^ ^{2} - 1}]$ . We call a sequence $(I^µ_{i})_{i = 1}^{\infty} \in {0, 1}^{N}$ a $I^^{2}$ -expansion for $x$ if $x=\sum_{i=1}^{\infty}Îµ_{i}Î²^{-i}$.…

math.NT2025

A quantitative framework for sets of exact approximation order by rational numbers

Simon Baker, Benjamin Ward

In this paper we study a quantitative notion of exactness within Diophantine approximation. Given $I^¨ : (0, \infty) \to (0, \infty)$ and $I ¨ : (0, \infty) \to (0, 1)$ satisfying $\lim_{q\to\…

math.NT2020

Equidistribution results for sequences of polynomials

Simon Baker

Let $(f_{n})_{n = 1}^{\infty}$ be a sequence of polynomials and $I^\pm > 1$ . In this paper we study the distribution of the sequence $(f_{n} (I^\pm))_{n = 1}^{\infty}$ modulo one. We give sufficien…

math.DS2021

Equidistribution results for self-similar measures

Simon Baker

A well known theorem due to Koksma states that for Lebesgue almost every $x > 1$ the sequence $(x^{n})_{n = 1}^{\infty}$ is uniformly distributed modulo one. In this paper we give suffic…

math.DS2025

On the Fourier transform of random Bernoulli convolutions

Simon Baker, Henna Koivusalo, Sascha Troscheit +1

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ Î¼_Ï= \mathop{\circledast}_{k=1}^{\infty} \left( \frac{Î´_0 + Î´_{Î…

math.DS2024

On the strong separation condition for self-similar iterated function systems with random translations

Simon Baker, Derong Kong, Zhiqiang Wang

Given a self-similar iterated function system $I^¦ = {I ¨ _{i} (x) = I ¨ _{i} O_{i} x + t_{i}}_{i = 1}^{m}$ acting on $R^{d}$ , we can generate a parameterised family of iterated function system…

math.NT2018

Bifurcation sets arising from non-integer base expansions

Pieter Allaart, Simon Baker, Derong Kong

Given a positive integer $M$ and $q \in (1, M + 1]$ , let $U_{q}$ 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\ldot…

cs.CL2024

Can Rule-Based Insights Enhance LLMs for Radiology Report Classification? Introducing the RadPrompt Methodology

Panagiotis Fytas, Anna Breger, Ian Selby +3

Developing imaging models capable of detecting pathologies from chest X-rays can be cost and time-prohibitive for large datasets as it requires supervision to attain state-of-the-a…

math.DS2024

Quantitative recurrence and the shrinking target problem for overlapping iterated function systems

Simon Baker, Henna Koivusalo

In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function syst…

math.DS2025

Polynomial Fourier decay for fractal measures and their pushforwards

Simon Baker, Amlan Banaji

We prove that the pushforwards of a very general class of fractal measures $I^¼$ on $R^{d}$ under a large family of non-linear maps $F : R^{d} \to R$ exh…

math.DS2017

Exceptional digit frequencies and expansions in non-integer bases

Simon Baker

In this paper we study the set of digit frequencies that are realised by elements of the set of $I^^{2}$ -expansions. The main result of this paper demonstrates that as $I^^{2}$ approaches…

math.DS2026

Self-similar and self-conformal measures with slow Fourier decay

Simon Baker, Amlan Banaji

Given any function $I ¨ : [0, \infty) \to (0, 1]$ satisfying $lim_{I^¾ \to \infty} I ¨ (I^¾) = 0$ , we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-…

math.NT2014

On the distribution of powers of real numbers modulo 1

Simon Baker

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 s…

math.DS2013

On small bases which admit countably many expansions

Simon Baker

Let $q \in (1, 2)$ and $x \in [0, \frac{1}{q - 1}]$ . We say that a sequence $(I^µ_{i})_{i = 1}^{\infty} \in {0, 1}^{N}$ is an expansion of $x$ in base $q$ (or a $q$ -expansion) if x=\sum_{…

math.DS2020

Iterated function systems with super-exponentially close cylinders II

Simon Baker

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $R$ with no exact overlaps for…

math.DS2012

The growth rate and dimension theory of beta-expansions

Simon Baker

In a recent paper of Feng and Sidorov they show that for $I^^{2} \in (1, \frac{1 + 5}{2})$ the set of $I^^{2}$ -expansions grows exponentially for every $x \in (0, \frac{1}{I ^ ^{2} - 1})$ . In this…

math.NT2015

Approximation properties of $I^^{2}$ -expansions II

Simon Baker

Given $I^^{2} \in (1, 2)$ and $x \in [0, \frac{1}{I ^ ^{2} - 1}]$ , a sequence $(I^µ_{i})_{i = 1}^{\infty} \in {0, 1}^{N}$ is called a $I^^{2}$ -expansion for $x$ if $$x=\sum_{i=1}^{\infty}\frac{Îµ_{…

math.DS2014

Dynamical properties of S-gap shifts and other shift spaces

Simon Baker, Andrei E. Ghenciu

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…

math.DS2019

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

Simon Baker, Derong Kong

Given $I^^{2} \in (1, 2],$ the $I^^{2}$ -transformation $T_{I^}^{2} : x \mapsto I^^{2} x (mod 1)$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-…

math.DS2017

Numbers with simply normal $I^^{2}$ -expansions

Simon Baker, Derong Kong

In [Bak] the first author proved that for any $I^^{2} \in (1, I^_{K L}^{2})$ every $x \in (0, \frac{1}{I ^ ^{2} - 1})$ has a simply normal $I^^{2}$ -expansion, where $I^_{K L}^{2} \approx 1.78723$ is the Komornik-…

math.CA2015

Inhomogeneous self-similar sets with overlaps

Simon Baker, Jonathan M. Fraser, AndrÃ¡s MÃ¡thÃ©

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 th…

math.DS2016

Unique expansions and intersections of Cantor sets

Simon Baker, Derong Kong

To each $I^\pm \in (1/3, 1/2)$ we associate the Cantor set $I^_{I^} \pm := {i = 1 \sum \infty I^µ_{i} I^\pm^{i} : I^µ_{i} \in {0, 1}, i \geq 1} .$ In this paper we consider the intersection $Î_…

math.DS2018

Maximising Bernoulli measures and dimension gaps for countable branched systems

Simon Baker, Natalia Jurga

Kifer, Peres, and Weiss proved that there exists $c_{0} > 0,$ such that $dim I^¼ \leq 1 - c_{0}$ for any probability measure $I^¼$ which makes the digits of the continued fraction expansion…

math.DS2020

Analogues of Khintchine's theorem for random attractors

Simon Baker, Sascha Troscheit

In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine appr…

cs.CL2021

Few-Shot Table-to-Text Generation with Prototype Memory

Yixuan Su, Zaiqiao Meng, Simon Baker +1

Neural table-to-text generation models have achieved remarkable progress on an array of tasks. However, due to the data-hungry nature of neural models, their performances strongly…

math.DS2013

On universal and periodic $I^^{2}$ -expansions, and the Hausdorff dimension of the set of all expansions

Simon Baker

In this paper we study the topology of a set naturally arising from the study of $I^^{2}$ -expansions. After proving several elementary results for this set we study the case when our b…

math.DS2025

Disintegration results for fractal measures and applications to Diophantine approximation

Simon Baker

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self…

math.NT2022

A note on dyadic approximation in Cantor's set

Demi Allen, Simon Baker, Sam Chow +1

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, $K$ , for approximation functions of the form $I ¨ _{I ¨} (n) = n^{- I ¨ }$ ( $I ¨ \geq 0$ ). In partic…

math.DS2026

Pair correlation statistics for dynamical systems

Simon Baker, Mike Todd

We study the pair correlation statistics of orbits generated by maps on the interval. We show that under suitable mixing and multifractal assumptions, the pair correlation statisti…

math.DS2024

Fourier Decay from $L^{2}$ -Flattening

Simon Baker, Osama Khalil, Tuomas Sahlsten

We develop a unified approach for establishing rates of decay for the Fourier transform of a wide class of dynamically defined measures. Among the key features of the method is the…

math.NT2014

Expansions in non-integer bases: lower order revisited

Simon Baker, Nikita Sidorov

Let $q \in (1, 2)$ and $x \in [0, \frac{1}{q - 1}]$ . We say that a sequence $(ε_{i})_{i = 1}^{\infty} \in {0, 1}^{N}$ is an expansion of $x$ in base $q$ (or a $q$ -expansion) if…

math.NT2018

A General Mass Transference Principle

Demi Allen, Simon Baker

In this paper we prove a general form of the Mass Transference Principle for $lim sup$ sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such set…

math.DS2020

Quantitative recurrence properties for self-conformal sets

Simon Baker, Michael Farmer

In this paper we study the quantitative recurrence properties of self-conformal sets $X$ equipped with the map $T : X \to X$ induced by the left shift. In particular, given a function…

math.NT2022

Approximating elements of the middle third Cantor set with dyadic rationals

Simon Baker

Let $C$ be the middle third Cantor set and $I^¼$ be the $\frac{l o g 2}{l o g 3}$ -dimensional Hausdorff measure restricted to $C$ . In this paper we study approximations of elements of…

math.DS2019

Iterated function systems with super-exponentially close cylinders

Simon Baker

Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $R$ does not equal its expected value, then t…

math.DS2014

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

Simon Baker

Let $I^»_{1}, \dots, I^»_{n}$ be real numbers in $(0, 1)$ and $p_{1}, \dots, p_{n}$ be points in $R^{d}$ . Consider the collection of maps $f_{j}:\mathbb{R}^{d}\to\mathbb{R}^{d}…

cs.CL2021

Dialogue Response Selection with Hierarchical Curriculum Learning

Yixuan Su, Deng Cai, Qingyu Zhou +6

We study the learning of a matching model for dialogue response selection. Motivated by the recent finding that models trained with random negative samples are not ideal in real-wo…

math.NT2015

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

Yuru Zou, Lijin Wang, Jian Lu +1

Let $q \in (1, 2)$ . A $q$ -expansion of a number $x$ in $[0, \frac{1}{q - 1}]$ is a sequence $(I^´_{i})_{i = 1}^{\infty} \in {0, 1}^{N}$ satisfying $$ x=\sum_{i=1}^\infty\frac{Î´_i}{q^i}…

cs.CL2020

Multi-SimLex: A Large-Scale Evaluation of Multilingual and Cross-Lingual Lexical Semantic Similarity

Ivan VuliÄ, Simon Baker, Edoardo Maria Ponti +9

We introduce Multi-SimLex, a large-scale lexical resource and evaluation benchmark covering datasets for 12 typologically diverse languages, including major languages (e.g., Mandar…

math.DS2012

A multifractal zeta function for cookie cutter sets

Simon Baker

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…

math.NT2014

Badly approximable numbers for sequences of balls

Simon Baker

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 genera…

math.DS2012

Generalised golden ratios over integer alphabets

Simon Baker

It is a well known result that for $I^^{2} \in (1, \frac{1 + 5}{2})$ and $x \in (0, \frac{1}{I ^ ^{2} - 1})$ there exists uncountably many $(I^µ_{i})_{i = 1}^{\infty} \in 0, 1^{N}$ such t…

math.NT2020

Gap statistics and higher correlations for geometric progressions modulo one

Christoph Aistleitner, Simon Baker, Niclas Technau +1

Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $I^\pm > 1$ , the fractional parts of the geometric progression $(I^\pm^{n})_{n \geq 1}$ are equidistributed…

math.DS2020

Overlapping iterated function systems from the perspective of Metric Number Theory

Simon Baker

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approxim…

math.DS2015

Induced Random $I^^{2}$ -transformation

Simon Baker, Karma Dajani

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…

cs.CL2021

Non-Autoregressive Text Generation with Pre-trained Language Models

Yixuan Su, Deng Cai, Yan Wang +4

Non-autoregressive generation (NAG) has recently attracted great attention due to its fast inference speed. However, the generation quality of existing NAG models still lags behind…

math.DS2014

On univoque points for self-similar sets

Simon Baker, Karma Dajani, Kan Jiang

Let $K \subseteq R$ 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 codin…

math.DS2021

On normal numbers and self-similar measures

Simon Baker

In this paper we prove that if ${I ¨ _{i} (x) = I^» x + t_{i}}$ is an equicontractive iterated function system and $b$ is a positive integer satisfying $\frac{\log b}{\log |Î»|}\notin\mathbb{…

math.DS2017

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

Simon Baker

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…

Publications (69)

Recurrence rates for shifts of finite type

Distinct dimensions for attractors of bi-Lipschitz iterated function systems

On periodic representations in non-Pisot bases

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

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

On the regularity of the generalised golden ratio function

Metric results for numbers with multiple q-expansions

On normal numbers and self-similar measures

On finitely many base q expansions

Spectral gaps and Fourier dimension for self-conformal sets with overlaps

New dimension bounds for I^±I^2 sets

Stylistic Dialogue Generation via Information-Guided Reinforcement Learning Strategy

On the cardinality and dimension of the slices of Okamoto's functions

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

Rethinking Clinical Relevance in Chest X-ray Machine Learning: How Evaluation References Define Performance

An analogue of Khintchine's theorem for self-conformal sets

On the pair correlations of powers of real numbers

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

Prototype-to-Style: Dialogue Generation with Style-Aware Editing on Retrieval Memory

Intrinsic Diophantine Approximation for overlapping iterated function systems

Approximation properties of I^2-expansions

A quantitative framework for sets of exact approximation order by rational numbers

Equidistribution results for sequences of polynomials

Equidistribution results for self-similar measures

On the Fourier transform of random Bernoulli convolutions

On the strong separation condition for self-similar iterated function systems with random translations

Bifurcation sets arising from non-integer base expansions

Can Rule-Based Insights Enhance LLMs for Radiology Report Classification? Introducing the RadPrompt Methodology

Quantitative recurrence and the shrinking target problem for overlapping iterated function systems

Polynomial Fourier decay for fractal measures and their pushforwards

Exceptional digit frequencies and expansions in non-integer bases

Self-similar and self-conformal measures with slow Fourier decay

On the distribution of powers of real numbers modulo 1

On small bases which admit countably many expansions

Iterated function systems with super-exponentially close cylinders II

The growth rate and dimension theory of beta-expansions

Approximation properties of I^2-expansions II

Dynamical properties of S-gap shifts and other shift spaces

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

Numbers with simply normal I^2-expansions

Inhomogeneous self-similar sets with overlaps

Unique expansions and intersections of Cantor sets

Maximising Bernoulli measures and dimension gaps for countable branched systems

Analogues of Khintchine's theorem for random attractors

Few-Shot Table-to-Text Generation with Prototype Memory

On universal and periodic I^2-expansions, and the Hausdorff dimension of the set of all expansions

Disintegration results for fractal measures and applications to Diophantine approximation

A note on dyadic approximation in Cantor's set

Pair correlation statistics for dynamical systems

Fourier Decay from L2-Flattening

Expansions in non-integer bases: lower order revisited

A General Mass Transference Principle

Quantitative recurrence properties for self-conformal sets

Approximating elements of the middle third Cantor set with dyadic rationals

Iterated function systems with super-exponentially close cylinders

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

Dialogue Response Selection with Hierarchical Curriculum Learning

On small bases for which 1 has countably many expansions

Multi-SimLex: A Large-Scale Evaluation of Multilingual and Cross-Lingual Lexical Semantic Similarity

A multifractal zeta function for cookie cutter sets

Badly approximable numbers for sequences of balls

Generalised golden ratios over integer alphabets

Gap statistics and higher correlations for geometric progressions modulo one

Overlapping iterated function systems from the perspective of Metric Number Theory

Induced Random I^2-transformation

Non-Autoregressive Text Generation with Pre-trained Language Models

On univoque points for self-similar sets

On normal numbers and self-similar measures

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

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

Metric results for numbers with multiple $q$ -expansions

On finitely many base $q$ expansions

New dimension bounds for $I^\pm I^^{2}$ sets

Approximation properties of $I^^{2}$ -expansions

Approximation properties of $I^^{2}$ -expansions II

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

Numbers with simply normal $I^^{2}$ -expansions

On universal and periodic $I^^{2}$ -expansions, and the Hausdorff dimension of the set of all expansions

Fourier Decay from $L^{2}$ -Flattening

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

Induced Random $I^^{2}$ -transformation