P. Allaart

activity

20062023

most citedOn the continuity of the Hausdorff dimension of the univoque set

17 citations · 59 across the 22 of their papers we have counts for

collaborators

31 papers

math.PR2023

Random subsets of Cantor sets generated by trees of coin flips

Pieter Allaart, Taylor Jones

We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0, 1]$ via random labelings of an infinite $M$ -ary tree, where $M \geq 2$ . The Cantor set…

math.DS2023

Entropy plateaus, transitivity and bifurcation sets for the $β$ -transformation with a hole at $0$

Pieter Allaart, Derong Kong

Given $β > 1$ , let $T_{β}$ be the $β$ -transformation on the unit circle $[0, 1)$ such that $T_{β} (x) = β x (mod 1)$ . For each $t \in [0, 1)$ let $K_{β} (t)$ be the survivor set consisting of all $x…

math.CA2022

Box-counting dimension and differentiability of box-like statistically self-affine functions

Pieter Allaart, Taylor Jones

We consider a class of "box-like" statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiab…

math.DS2021

Critical values for the $β$ -transformation with a hole at $0$

Pieter Allaart, Derong Kong

Given $β \in (1, 2]$ , let $T_{β}$ be the $β$ -transformation on the unit circle $[0, 1)$ such that $T_{β} (x) = β x (mod 1)$ . For each $t \in [0, 1)$ let $K_{β} (t)$ be the survivor set consisting of…

math.NT2021

On the existence of numbers with matching continued fraction and decimal expansions

Pieter Allaart, Stephen Jackson, Taylor Jones +1

A Trott number is a number $x \in (0, 1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$ , in the following sense: If $x = [0; a_{1}, a_{2}, \dots]$ ,…

math.DS2020

Density spectrum of Cantor measure

Pieter Allaart, Derong Kong

Given $ρ \in (0, 1/3]$ , let $μ$ be the Cantor measure satisfying $μ = \frac{1}{2} μ f_{0}^{- 1} + \frac{1}{2} μ f_{1}^{- 1}$ , where $f_{i} (x) = ρ x + i (1 - ρ)$ for $i = 0, 1$ . The support of $μ$ is a Cantor…

math.NT2020★ 1 cited

On the smallest base in which a number has a unique expansion

Pieter Allaart, Derong Kong

Given a real number $x > 0$ , we determine $q_{s} (x) := in f U (x)$ , where $U (x)$ is the set of all bases $q \in (1, 2]$ for which $x$ has a unique expansion of $0$ 's and $…

math.CA2019★ 1 cited

The pointwise Hölder spectrum of general self-affine functions on an interval

Pieter Allaart

This paper gives the pointwise Hölder (or multifractal) spectrum of continuous functions on the interval $[0, 1]$ whose graph is the attractor of an iterated function system consist…

math.DS2018

An algebraic approach to entropy plateaus in non-integer base expansions

Pieter C. Allaart

For a positive integer $M$ and a real base $q \in (1, M + 1]$ , let $U_{q}$ denote the set of numbers having a unique expansion in base $q$ over the alphabet ${0, 1, \dots, M}$ , a…

math.PR2018

A random walk version of Robbins' problem: small horizon

Pieter C. Allaart, Andrew Allen

In Robbins' problem of minimizing the expected rank, a finite sequence of $n$ independent, identically distributed random variables are observed sequentially and the objective is t…

math.DS2018★ 9 cited

Relative bifurcation sets and the local dimension of univoque bases

Pieter Allaart, Derong Kong

Fix an alphabet $A = {0, 1, \dots, M}$ with $M \in N$ . The univoque set $U$ of bases $q \in (1, M + 1)$ in which the number $1$ has a unique expansion over the alphabet $A…

math.DS2018★ 17 cited

On the continuity of the Hausdorff dimension of the univoque set

Pieter Allaart, Derong Kong

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $U_{q}$ of numbers having a uniq…

math.CA2017

Differentiability and Hölder spectra of a class of self-affine functions

Pieter C. Allaart

This paper studies a large class of continuous functions $f : [0, 1] \to R^{d}$ whose range is the attractor of an iterated function system ${S_{1}, \dots, S_{m}}$ consisting of si…

math.NT2017★ 6 cited

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…

math.CA2016

Differentiability of a two-parameter family of self-affine functions

Pieter C. Allaart

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\em $β$ -expansions}) and the infinite derivatives of a class of se…

math.NT2016

On univoque and strongly univoque sets

Pieter C. Allaart

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet ${0, 1, \dots, α}$ and a real number (base) $1 < β < α + 1$ , the so-called…

math.PR2015

A sharp lower bound for choosing the maximum of an independent sequence

Pieter C. Allaart, Jose A. Islas

This paper considers a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The…

math.CA2015

The infinite derivatives of Okamoto's self-affine functions: an application of beta-expansions

Pieter C. Allaart

Okamoto's one-parameter family of self-affine functions $F_{a} : [0, 1] \to [0, 1]$ , where $0 < a < 1$ , includes the continuous nowhere differentiable functions of Perkins ( $a = 5/6$ ) and Bourb…

math.CA2013

On the level sets of the Takagi-van der Waerden functions

Pieter C. Allaart

This paper examines the level sets of the continuous but nowhere differentiable functions \begin{equation*} f_r(x)=\sum_{n=0}^\infty r^{-n}ϕ(r^n x), \end{equation*} where $ϕ (x)$ is…

math.CA2013

Correction and strengthening of "How large are the level sets of the Takagi function?"

Pieter C. Allaart

The purpose of this note is to correct an error in an earlier paper by the author about the level sets of the Takagi function [Monatsh. Math. 167 (2012), 311-331 and arXiv:1102.161…

math.CA2013★ 1 cited

Hausdorff dimension of level sets of generalized Takagi functions

Pieter C. Allaart

This paper examines level sets of two families of continuous, nowhere differentiable functions (one a subfamily of the other) defined in terms of the "tent map". The well-known Tak…

math.CA2012

Level sets of signed Takagi functions

Pieter C. Allaart

This paper examines level sets of functions of the form $f (x) = \sum_{n = 0}^{\infty} \frac{r _{n}}{2 ^{n}} ϕ (2^{n} x)$ , where phi(x) is the distance from x to the nearest integer, and r_n equals…

math.NT2012

Digital sum inequalities and approximate convexity of Takagi-type functions

Pieter C. Allaart

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of…

math.CA2011★ 6 cited

The Takagi function: a survey

Pieter Allaart, Kiko Kawamura

This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and…

Random subsets of Cantor sets generated by trees of coin flips

Entropy plateaus, transitivity and bifurcation sets for the β-transformation with a hole at 0

Box-counting dimension and differentiability of box-like statistically self-affine functions

Critical values for the β-transformation with a hole at 0

On the existence of numbers with matching continued fraction and decimal expansions

Density spectrum of Cantor measure

On the smallest base in which a number has a unique expansion

The pointwise Hölder spectrum of general self-affine functions on an interval

An algebraic approach to entropy plateaus in non-integer base expansions

A random walk version of Robbins' problem: small horizon

Relative bifurcation sets and the local dimension of univoque bases

On the continuity of the Hausdorff dimension of the univoque set

Differentiability and Hölder spectra of a class of self-affine functions

Bifurcation sets arising from non-integer base expansions

Differentiability of a two-parameter family of self-affine functions

On univoque and strongly univoque sets

A sharp lower bound for choosing the maximum of an independent sequence

The infinite derivatives of Okamoto's self-affine functions: an application of beta-expansions

On the level sets of the Takagi-van der Waerden functions

Correction and strengthening of "How large are the level sets of the Takagi function?"

Hausdorff dimension of level sets of generalized Takagi functions

Level sets of signed Takagi functions

Digital sum inequalities and approximate convexity of Takagi-type functions

The Takagi function: a survey

Entropy plateaus, transitivity and bifurcation sets for the $β$ -transformation with a hole at $0$

Critical values for the $β$ -transformation with a hole at $0$