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James D. Currie

Publications and source records attributed to James D. Currie.

At least 19 recordsLinked to original sources
arXiv2026

Words with factor complexity $2n+1$ and minimal critical exponent

Word ${\mathbf G}$ is the fixed point of the morphism $γ=[01,2,02]$. In 2019, Shallit and Shur showed that ${\mathbf G}$ has factor complexity $2n+1$. They also showed that ${\mathbf G}$ has critical exponent $μ=2+\frac{1}{λ^2-1}= 2.4808726\cdots$, where $λ=1.7548777$ is the real zero of $x^3-2x+x-1=0$. They conjectured that this was the least possible critical exponent among words with factor complexity $2n+1$. We confirm their conjecture. The proof, using an intricate case analysis, is by computer. The relevant program generates a `human readable' proof.

math.CO↗
arXiv2025

The repetition threshold for ternary rich words

In 2017, Vesti proposed the problem of determining the repetition threshold for infinite rich words, i.e., for infinite words in which all factors of length $n$ contain $n$ distinct nonempty palindromic factors. In 2020, Currie, Mol, and Rampersad proved a conjecture of Baranwal and Shallit that the repetition threshold for binary rich words is $2 + \sqrt{2}/2$. In this paper, we prove a structure theorem for $16/7$-power-free ternary rich words. Using the structure theorem, we deduce that the repetition threshold for ternary rich words is $1 + 1/(3 - μ) \approx 2.25876324$, where $μ$ is the unique real root of the polynomial $x^3 - 2x^2 - 1$.

math.CO↗
arXiv2023

A small morphism giving Abelian repetition threshold less than 2

It is known that there are infinite words over finite alphabets with Abelian repetition threshold arbitrarily close to 1; however, the construction previously used involves huge alphabets. In this note we give a short cyclic morphism (length 13) over an 8-letter alphabet yielding an Abelian repetition threshold less than 1.8.

math.CO↗
arXiv2023

The analogue of overlap-freeness for the Fibonacci morphism

A $4^-$-power is a non-empty word of the form $XXXX^-$, where $X^-$ is obtained from $X$ by erasing the last letter. A binary word is called {\em faux-bonacci} if it contains no $4^-$-powers, and no factor 11. We show that faux-bonacci words bear the same relationship to the Fibonacci morphism that overlap-free words bear to the Thue-Morse morphism. We prove the analogue of Fife's Theorem for faux-bonacci words, and characterize the lexicographically least and greatest infinite faux-bonacci words.

math.CO↗
arXiv2023

The analogue of overlap-freeness for the period-doubling sequence

Good words are binary words avoiding factors 11 and 1001, and patterns 0000 and 00010100. We show that good words bear the same relationship to the period-doubling sequence that overlap-free words bear to the Thue-Morse sequence. We prove an analogue of Fife's Theorem for good words, exhibit the lexicographically least and greatest infinite good words, and determine the patterns avoided by the period doubling word.

math.CO↗
arXiv2020

The undirected repetition threshold and undirected pattern avoidance

For a rational number $r$ such that $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where the word $x$ is nonempty, the word $x'$ is in $\{x,x^R\}$, and we have $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mbox{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mbox{URT}(3)=\tfrac{7}{4}$. Then we show that $\mbox{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mbox{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conjecture for $k\in\{4,5,\ldots,21\}.$ We then consider related problems in pattern avoidance; in particular, we find the undirected avoidability index of every binary pattern. This is an extended version of a paper presented at WORDS 2019, and it contains new and improved results.

math.CO↗
arXiv2020

The repetition threshold for binary rich words

A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that the repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article, we give a structure theorem for infinite binary rich words that avoid $14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence, we deduce that the repetition threshold for binary rich words is $2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open problem of Vesti for the binary alphabet; the problem remains open for larger alphabets.

math.CO↗
arXiv2019

The undirected repetition threshold

For rational $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where $x$ is nonempty, $x'\in\{x,x^\mathrm{R}\}$, and $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mathrm{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mathrm{URT}(3)=\tfrac{7}{4}$. Then we show that $\mathrm{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mathrm{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conjecture for $k\in\{4,8,12\}.$

math.CO↗
arXiv2018

Circular repetition thresholds on some small alphabets: Last cases of Gorbunova's conjecture

A word is called $β$-free if it has no factors of exponent greater than or equal to $β$. The repetition threshold $\mathrm{RT}(k)$ is the infimum of the set of all $β$ such that there are arbitrarily long $k$-ary $β$-free words (or equivalently, there are $k$-ary $β$-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all $β$ such that - there are arbitrarily long $k$-ary $β$-free circular words is called the weak circular repetition threshold, denoted $\mathrm{CRT}_{\mathrm{W}}(k)$; - there are $k$-ary $β$-free circular words of every sufficiently large length is called the intermediate circular repetition threshold, denoted $\mathrm{CRT}_{\mathrm{I}}(k)$; - there are $k$-ary $β$-free circular words of every length is called the strong circular repetition threshold, denoted $\mathrm{CRT}_{\mathrm{S}}(k)$. We prove that $\mathrm{CRT}_{\mathrm{S}}(4)=\tfrac{3}{2}$ and $\mathrm{CRT}_{\mathrm{S}}(5)=\tfrac{4}{3}$, confirming a conjecture of Gorbunova and providing the last unknown values of the strong circular repetition threshold. We also prove that $\mathrm{CRT}_{\mathrm{I}}(3)=\mathrm{CRT}_{\mathrm{W}}(3)=\mathrm{RT}(3)=\tfrac{7}{4}$.

math.CO↗
arXiv2016

Chromatic properties of the Euclidean plane

Let $G$ be the unit distance graph in the plane. A well-known problem in combinatorial geometry is that of determining the chromatic number of $G$. It is known that $4\le χ(G)\le 7$. The upper bound of 7 is obtained using tilings of the plane. The present paper studies two problems where we seek proper colourings of $G$, adding restrictions inspired by tilings: Let $H(ε)$ be the graph whose vertices are the points of ${\mathbb R}^2$, with an edge between two points if their distance lies in the interval $[1,1+ε]$. We show that for small $ε$, $0<ε\le \frac{3\sqrt{2}}{4}-1$, we have $6\le χ(H(ε))\le 7$. This improves the result of Exoo and Grytczuk et al. that $5\le χ(H(ε))$ for small $ε$. Suppose that $G$ is properly coloured, but so that two solidly coloured regions meet along a straight line in some neighbourhood. Then at least 5 colours must be used.

math.CO↗
arXiv2016

A ternary square-free sequence avoiding factors equivalent to $abcacba$

We solve a problem of Petrova, finalizing the classification of letter patterns avoidable by ternary square-free words; we show that there is a ternary square-free word avoiding letter pattern $xyzxzyx$. In fact, we: (1) characterize all the (two-way) infinite ternary square-free words avoiding letter pattern $xyzxzyx$ (2) characterize the lexicographically least (one-way) infinite ternary square-free word avoiding letter pattern $xyzxzyx$ (3) show that the number of ternary square-free words of length $n$ avoiding letter pattern $xyzxzyx$ grows exponentially with $n$.

cs.FL↗
arXiv2015

Binary words avoiding xx^Rx and strongly unimodal sequences

In previous work, Currie and Rampersad showed that the growth of the number of binary words avoiding the pattern xxx^R was intermediate between polynomial and exponential. We now show that the same holds for the growth of the number of binary words avoiding the pattern xx^Rx. Curiously, the analysis for xx^Rx is much simpler than that for xxx^R. We derive our results by giving a bijection between the set of binary words avoiding xx^Rx and a class of sequences closely related to the class of "strongly unimodal sequences."

math.CO↗
arXiv2015

Growth rate of binary words avoiding $xxx^R$

Consider the set of those binary words with no non-empty factors of the form $xxx^R$. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the existence of upper and lower bounds on the number of such words of length $n$, where each of these bounds is asymptotically equivalent to a (different) function of the form $Cn^{\lg n+c}$, where $C$, $c$ are constants.

cs.FL↗