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Publications (45)

math.NT2011

On the multiplicative ErdÅs discrepancy problem

Michael Coons

As early as the 1930s, PÃ¡l ErdÅs conjectured that: {\em for any multiplicative function $f : N \to {- 1, 1}$ , the partial sums $\sum_{n \leq x} f (n)$ are unbounded.} Consideri…

math.CO2024

Relative position in binary substitutions

Michael Coons, Christopher Ramsey, Nicolae Strungaru

Given an infinite word $w$ on a finite alphabet, an immediate question arises:~can we understand the frequency of letters in $w$ \,? For words that are the fixed points…

math.NT2008

Completely multiplicative functions taking values in ${- 1, 1}$

Peter Borwein, Stephen K. K. Choi, Michael Coons

Define {\em the Liouville function for $A$ }, a subset of the primes $P$ , by $I^»_{A} (n) = (- 1)^{I^c ◯_{A} (n)}$ where $I^c ◯_{A} (n)$ is the number of prime factors of $n$ coming from $A$ counti…

math.NT2008

Transcendence of Power Series for Some Number Theoretic Functions

Michael Coons, Peter Borwein

We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This res…

math.NT2021

The spectral theory of regular sequences

Michael Coons, James Evans, Neil Manibo

Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of…

math.NT2020

A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension

Michael Coons, James Evans

We show that missing $q$ -ary digit sets $F \subseteq [0, 1]$ have corresponding naturally associated countable binary $q$ -automatic sequence $f$ . Using this correspondence, we show th…

math.NT2016

A dichotomy law for the Diophantine properties in $I^^{2}$ -dynamical systems

Michael Coons, Mumtaz Hussain, Bao-Wei Wang

Let $I^^{2} > 1$ be a real number and define the $I^^{2}$ -transformation on $[0, 1]$ by $T_{I^}^{2} : x \mapsto I^^{2} x mod 1$ . Further, define $$W_y(T_Î²,Î¨):=\{x\in [0, 1]:|T_Î²^nx-y|<Î¨(n) \mbox{ for…

math.CO2015

Regular sequences and the joint spectral radius

Michael Coons

We classify the growth of a $k$ -regular sequence based on information from its $k$ -kernel. In order to provide such a classification, we introduce the notion of a growth exponent f…

math.NT2010

On some conjectures concerning Stern's sequence and its twist

Michael Coons

In a recent paper, Roland Bacher conjectured three identities concerning Stern's sequence and its twist. In this paper we prove Bacher's conjectures. Possibly of independent intere…

math.NT2018

A natural probability measure derived from Stern's diatomic sequence

Michael Baake, Michael Coons

Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit in…

math.DS2020

Binary constant-length substitutions and Mahler measures of Borwein polynomials

Michael Baake, Michael Coons, Neil Manibo

We show that the Mahler measure of every Borwein polynomial -- a polynomial with coefficients in ${- 1, 0, 1}$ having non-zero constant term -- can be expressed as a maximal Lyapu…

math.NT2021

Ghost distributions of regular sequences are affine transformations of self-affine sets

Michael Coons, James Evans, Zachary Groth +1

Ghost measures of regular sequences---the unbounded analogue of automatic sequences---are generalisations of standard fractal mass distributions. They were introduced to determine…

math.NT2020

Scaling of the diffraction measure of $k$ -free integers near the origin

Michael Baake, Michael Coons

Asymptotics are derived for the scaling of the total diffraction intensity for the set of $k$ -free integers near the origin, which is a measure for the degree of patch fluctuations…

math.NT2010

Transcendence of generating functions whose coefficients are multiplicative

Jason P. Bell, Nils Bruin, Michael Coons

In this paper, we give a new proof and an extension of the following result of BÃ©zivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteris…

math.DS2023

Correlations of the Thue--Morse sequence

Michael Baake, Michael Coons

The pair correlations of the Thue--Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations…

math.NT2008

(Non)Automaticity of number theoretic functions

Michael Coons

Denote by $I^» (n)$ Liouville's function concerning the parity of the number of prime divisors of $n$ . Using a theorem of Allouche, MendÃ¨s France, and PeyriÃ¨re and many classical r…

math-ph2008

General moment theorems for non-distinct unrestricted partitions

Michael Coons, Klaus Kirsten

A well-known result from Hardy and Ramanujan gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers. In this vein, we c…

math.CO2017

Proof of Northshield's conjecture concerning an analogue of Stern's sequence for $Z [2]$

Michael Coons

We prove a conjecture of Northshield by determining the maximal order of his analogue of Stern's sequence for $Z [2]$ . In particular, if $b$ is Northshield's analogu…

math.CO2015

The maximal order of hyper-( $b$ -ary)-expansions

Michael Coons, Lukas Spiegelhofer

Using methods developed by Coons and Tyler, we give a new proof of a recent result of Defant, by determining the maximal order of the number of hyper-( $b$ -ary)-expansions of a nonn…

math.DS2021

On a family of singular continuous measures related to the doubling map

Michael Baake, Michael Coons, James Evans +1

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely…

math.NT2017

Mahler takes a regular view of Zaremba

Michael Coons

In the theory of continued fractions, Zaremba's conjecture states that there is a positive integer $M$ such that each integer is the denominator of a convergent of an ordinary cont…

math.NT2011

A Pattern Sequence Approach to Stern's Sequence

Michael Coons, Jeffrey Shallit

Let w be a binary string and let a_w (n) be the number of occurrences of the word w in the binary expansion of n. As usual we let s(n) denote the Stern sequence; that is, s(0)=0, s…

math.NT2009

On the residue class distribution of the number of prime divisors of an integer

Michael Coons, Sander R. Dahmen

The {\em Liouville function} is defined by $\gl(n):=(-1)^{Î©(n)}$ where $I^c ◯ (n)$ is the number of prime divisors of $n$ counting multiplicity. Let $\z_m:=e^{2Ïi/m}$ be a primitive…

math.NT2023

Spectral theory of regular sequences: parametrisation and spectral characterisation

Michael Coons, James Evans, Philipp Gohlke +1

We extend the existence of ghost measures beyond nonnegative primitive regular sequences to a large class of nonnegative real-valued regular sequences. In the general case, where t…

math.NT2011

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Michael Coons

Let $\C{G}(z):=\sum_{n=0}^\infty z^{2^n}(1-z^{2^n})^{-1}$ denote the generating function of the ruler function, and $\C{F}(z):=\sum_{n=0}^\infty z^{2^n}(1+z^{2^n})^{-1}$; note that…

math.NT2018

Becker's conjecture on Mahler functions

Jason Bell, Frederic Chyzak, Michael Coons +1

In 1994, Becker conjectured that if $F (z)$ is a $k$ -regular power series, then there exists a $k$ -regular rational function $R (z)$ such that $F (z) / R (z)$ satisfies a Mahler-type fun…

math.NT2013

An arithmetical excursion via Stoneham numbers

Michael Coons

Let $p$ be a prime and $b$ a primitive root of $p^{2}$ . In this paper, we give an explicit formula for the number of times a value in $0, 1, ..., b - 1$ occurs in the periodic part of t…

math.NT2015

Algebraic independence of Mahler functions via radial asymptotics

Richard P. Brent, Michael Coons, Wadim Zudilin

We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behaviour of a Mahler function $f (z)$ as…

math.NT2017

An asymptotic approach in Mahler's method

Michael Coons

We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not…

math.NT2016

Zero order estimates for Mahler functions

Michael Coons

We give an upper bound for the zero order of the difference between a Mahler function and an algebraic function. This complements estimates of Nesterenko, Nishioka, and TÃ¶pfer, am…

math.NT2010

Transcendence of generating functions whose coefficients are multiplicative

Jason P. Bell, Michael Coons

Let $K$ be a field of characteristic 0, $f : N \to K$ be a multiplicative function, and $F (z) = \sum_{n \geq 1} f (n) z^{n} \in K [[z]]$ be algebraic over $K (z)$ . Then either there is…

math.NT2023

Linear independence of series related to the Thue--Morse sequence along powers

Michael Coons, Yohei Tachiya

The Thue--Morse sequence ${t (n)}_{n ⩾ 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$ , where $t (n) = 1$ (r…

math.NT2014

Growth degree classification for finitely generated semigroups of integer matrices

Jason P. Bell, Michael Coons, Kevin G. Hare

Let $A$ be a finite set of $d \times d$ matrices with integer entries and let $m_{n} (A)$ be the maximum norm of a product of $n$ elements of $A$ . In this…

math.NT2014

The minimal growth of a $k$ -regular sequence

Jason P. Bell, Michael Coons, Kevin G. Hare

We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to\mathbb{Z}$ is an unbounded $k$…

math.NT2014

The maximal order of Stern's diatomic sequence

Michael Coons, Jason Tyler

We answer a question of Calkin and Wilf concerning the maximal order of Stern's diatomic sequence. Specifically, we prove that $$\limsup_{n\to\infty}\frac{a(n)}{Ï^{\log_2 n}}=\fra…

math.NT2015

Diophantine approximation of Mahler numbers

Jason Bell, Yann Bugeaud, Michael Coons

Suppose that $F (x) \in Z [[x]]$ is a Mahler function and that $1/ b$ is in the radius of convergence of $F (x)$ . In this paper, we consider the approximation of $F (1/ b)$ by alg…

math.NT2025

On the absolute value of the autocorrelations of the Thue-Morse sequence

Michael Coons, Jan MazÃ¡Ä, Ari Pincus-Kazmar +1

Recently, Baake and Coons proved several results on the average size of the autocorrelations of the Thue--Morse sequence. They also considered the absolute value of the autocorrela…

math.NT2014

Towards the (ir)rationality of values of Dirichlet series

Michael Coons, Daniel Sutherland

We show that if $F (s)$ is a nondegenerate ordinary Dirichlet series with nonnegative coefficients and $F (k)$ is a rational number for all large enough positive integers $k$ , then t…

math.NT2025

Asymptotics for partitions over the Fibonacci numbers and related sequences

Michael Coons, Simon Kristensen, Mathias L. Laursen

In this paper, harkening back to ideas of Hardy and Ramanujan, Mahler and de Bruijn, with the addition of more recent results on the Fibonacci Dirichlet series, we determine the as…

math.NT2017

Extension of a theorem of Duffin and Schaeffer

Michael Coons

Let $r_{1}, \dots, r_{s} : Z_{n ⩾ 0} \to C$ be linearly recurrent sequences whose associated eigenvalues have arguments in $I ¨ Q$ and let $F(z):=\sum_{n\geq…

math.NT2015

Transcendence tests for Mahler functions

Jason P. Bell, Michael Coons

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue $I^»_{F}$ of a Mahler function $F (z)$ , and develop a quick test for t…

math.NT2015

Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers

Michael Coons

As a corollary of the main result of our recent paper, {\em On the rational approximation of the sum of the reciprocals of the Fermat numbers} published in this same journal, we pr…

math.NT2008

Transcendence of the Gaussian Liouville number and relatives

Peter Borwein, Michael Coons

{\em The Liouville number}, denoted $l$ , is defined by $l := 0.100101011101101111100...,$ where the $n$ th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function fo…

math.NT2015

Strong normality and generalised Copeland--ErdÅs numbers

Elliot Catt, Michael Coons, Jordan Velich

We prove that an infinite class of Copeland-ErdÅs numbers are not strongly normal and provide the analogous result for Bugeaud's Mahler-inspired extension of the Copeland-ErdÅs n…

math.CV2012

The rational-transcendental dichotomy of Mahler functions

Jason P. Bell, Michael Coons, Eric Rowland

In this paper, we give a new proof of a result due to Bezivin that a D-finite Mahler function is necessarily rational. This also gives a new proof of the rational-transcendental di…

Publications (45)

On the multiplicative ErdÅs discrepancy problem

Relative position in binary substitutions

Completely multiplicative functions taking values in {−1,1}

Transcendence of Power Series for Some Number Theoretic Functions

The spectral theory of regular sequences

A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension

A dichotomy law for the Diophantine properties in I^2-dynamical systems

Regular sequences and the joint spectral radius

On some conjectures concerning Stern's sequence and its twist

A natural probability measure derived from Stern's diatomic sequence

Binary constant-length substitutions and Mahler measures of Borwein polynomials

Ghost distributions of regular sequences are affine transformations of self-affine sets

Scaling of the diffraction measure of k-free integers near the origin

Transcendence of generating functions whose coefficients are multiplicative

Correlations of the Thue--Morse sequence

(Non)Automaticity of number theoretic functions

General moment theorems for non-distinct unrestricted partitions

Proof of Northshield's conjecture concerning an analogue of Stern's sequence for Z[2​]

The maximal order of hyper-(b-ary)-expansions

On a family of singular continuous measures related to the doubling map

Mahler takes a regular view of Zaremba

A Pattern Sequence Approach to Stern's Sequence

On the residue class distribution of the number of prime divisors of an integer

Spectral theory of regular sequences: parametrisation and spectral characterisation

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Becker's conjecture on Mahler functions

An arithmetical excursion via Stoneham numbers

Algebraic independence of Mahler functions via radial asymptotics

An asymptotic approach in Mahler's method

Zero order estimates for Mahler functions

Transcendence of generating functions whose coefficients are multiplicative

Linear independence of series related to the Thue--Morse sequence along powers

Growth degree classification for finitely generated semigroups of integer matrices

The minimal growth of a k-regular sequence

The maximal order of Stern's diatomic sequence

Diophantine approximation of Mahler numbers

On the absolute value of the autocorrelations of the Thue-Morse sequence

Towards the (ir)rationality of values of Dirichlet series

Asymptotics for partitions over the Fibonacci numbers and related sequences

Extension of a theorem of Duffin and Schaeffer

Transcendence tests for Mahler functions

Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers

Transcendence of the Gaussian Liouville number and relatives

Strong normality and generalised Copeland--ErdÅs numbers

The rational-transcendental dichotomy of Mahler functions

On the multiplicative ErdÅs discrepancy problem

Completely multiplicative functions taking values in ${- 1, 1}$

A dichotomy law for the Diophantine properties in $I^^{2}$ -dynamical systems

Scaling of the diffraction measure of $k$ -free integers near the origin

Proof of Northshield's conjecture concerning an analogue of Stern's sequence for $Z [2]$

The maximal order of hyper-( $b$ -ary)-expansions

The minimal growth of a $k$ -regular sequence

Strong normality and generalised Copeland--ErdÅs numbers