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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 asymptotic number of ways $p_F(n)$ to write an integer as the sum of non-distinct Fibonacci numbers. This appears to be the first such asymptotic result concerning non-distinct partitions over Fibonacci numbers. As well, under weak conditions, we prove analogous results for a general linear recurrences., Comment: 17 Pages. V2 does not address distinct partitions--there was a nuanced (fatal) flaw in that part of the paper

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E.~Miyanohara by showing that, for a fixed Pisot or Salem number $\beta>\sqrt{\varphi}=1.272019649\ldots$, the set of the numbers $$ 1,\quad \sum_{n\geqslant 1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant 1}\frac{t(n^2)}{\beta^{n}},\quad \dots, \quad \sum_{n\geqslant 1}\frac{t(n^k)}{\beta^{n}},\quad \dots $$ is linearly independent over the field $\mathbb{Q}(\beta)$, where $\varphi:=(1+\sqrt{5})/2$ is the golden ratio. Our result implies that for any $k\geqslant 1$ and for any $a_1,a_2,\ldots,a_k\in\mathbb{Q}(\beta)$, not all zero, the sequence \{$a_1t(n)+a_2t(n^2)+\cdots+a_kt(n^k)\}_{n\geqslant 1}$ cannot be eventually periodic., Comment: 9 pages

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 the ghost measure is not unique, we show that they can be parametrised by a compact abelian group. For a subclass of these measures, by replacing primitivity with a commutativity condition, we show that these measures have an infinite convolution structure similar to Bernoulli convolutions. Using this structure, we show that these ghost measures have pure spectral type. Further, we provide results towards a classification of the spectral type based on inequalities involving the spectral radius, joint spectral radius, and Lyapunov exponent of the underlying set of matrices. In the case that the underlying measure is pure point, we show that the support of the measure must be a subset of the rational numbers, a result that resolves a new case of the finiteness conjecture., Comment: 36 pages

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 of the Thue--Morse sequence with general real weights are effectively determined by a single value of the balanced $2$-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue--Morse sequence vanish, and that, for any even $n$, the $n$-point correlations of the balanced Thue--Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue--Morse system. We finish by showing how the correlations of the Thue--Morse system with general real weights can be derived from the balanced $2$-point correlations., Comment: 18 pages, revised version, with minor additions and improvements

Ghost measures of regular sequences---the unbounded analogue of automatic sequences---are generalisations of standard fractal mass distributions. They were introduced to determine fractal (or self-similar) properties of regular sequences similar to those related to automatic sequences. The existence and continuity of ghost measures for a large class of regular sequences was recently given by Coons, Evans and Ma\~nibo. In this paper, we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions---the distribution functions of ghost measures---of the above-mentioned class of regular sequences are sections of self-affine sets. As an application of our result, we show that the ghost distributions of the Zaremba sequences---regular sequences of the denominators of the convergents of badly approximable numbers---are all singular continuous., Comment: 15 pages

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 that the Hausdorff dimension of $F$ is equal to the base-$q$ logarithm of the Mahler eigenvalue of $f$. In addition, we demonstrate that the standard mass distribution $\nu_F$ supported on $F$ is equal to the ghost measure $\mu_f$ of $f$., Comment: 8 pages

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 measures associated with substitutions as motivation, we study the limiting asymptotics of regular sequences by constructing a systematic measure-theoretic framework surrounding them. The constructed measures are generalisations of mass distributions supported on attractors of iterated function systems., Comment: 19 pages

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 singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue--Morse measure., Comment: 15 pages, with illustrating examples and figures, revised and slightly expanded version

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., Comment: 8 pages, revised version with improved presentation

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 functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^\gamma Q(z)$ for some explicit non-negative integer $\gamma$ and such that $1/Q(z)$ is $k$-regular., Comment: 19 pages

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 Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer's problem for height-one polynomials having minimal Mahler measure becomes equivalent to a natural question from the spectral theory of binary constant-length substitutions. This supports another connection between Mahler measures and dynamics, beyond the well-known appearance of Mahler measures as entropies in algebraic dynamics., Comment: 19 pages, expository article, revised version with some improvements and additions

We prove a conjecture of Northshield by determining the maximal order of his analogue of Stern's sequence for $\mathbb{Z}[\sqrt{2}]$. In particular, if $b$ is Northshield's analogue, we prove that $$\limsup_{n\to\infty}\frac{2b(n)}{(2n)^{\log_3 (\sqrt{2}+1)}}=1.$$, Comment: 6 pages

Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where $f(n)\in\{r_1(n),\ldots,$ $r_s(n)\}$ for each $n\geqslant 0$. We prove that if $F(z)$ is bounded in a sector of its disk of convergence, it is a rational function. This extends a very recent result of Tang and Wang, who gave the analogous result when the sequence $f(n)$ takes on values of finitely many polynomials., Comment: 2 pages

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 interval. We construct this measure and show that it is purely singular continuous, with a strictly increasing, H\"older continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence., Comment: 13 pages, 3 figures; minor revision with some improvements and updates

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 continued fraction with partial quotients bounded by $M$. In this paper, to each such $M$ we associate a regular sequence---in the sense of Allouche and Shallit---and establish various properties and results concerning the generating function of the regular sequence. In particular, we determine the minimal algebraic relation concerning the generating function and its Mahler iterates., Comment: 12 pages, 2 figures

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 T\"opfer, among others, who considered polynomials evaluated at Mahler functions., Comment: 5 pages, version 2 has a few fixed typos

Let $\beta>1$ be a real number and define the $\beta$-transformation on $[0,1]$ by $T_\beta:x\mapsto \beta x\bmod 1$. Further, define $$W_y(T_{\beta},\Psi):=\{x\in [0, 1]:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\}$$ and $$W(T_{\beta},\Psi):=\{(x, y)\in [0, 1]^2:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\},$$ where $\Psi:\mathbb{N}\to\mathbb{R}_{>0}$ is a positive function such that $\Psi(n)\to 0$ as $n\to \infty$. In this paper, we show that each of the above sets obeys a Jarn\'ik-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets., Comment: Accepted for publication in Mathematika

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 nonnegative integer $n$ for general integral bases $b\geqslant 2$., Comment: 6 pages

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 prove that for each integer $b\geq 2$ the irrationality exponent of $\sum_{n\geqslant 0} s_2(n)/b^n$ is equal to $2$.

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 for $k$-regular sequences and show that this exponent is equal to the joint spectral radius of any set of a special class of matrices determined by the $k$-kernel., Comment: 5 pages

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue $\lambda_F$ of a Mahler function $F(z)$, and develop a quick test for the transcendence of $F(z)$ over $\mathbb{C}(z)$, which is determined by the value of the eigenvalue $\lambda_F$. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of $F(z)$. We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given., Comment: 9 pages

We prove that an infinite class of Copeland-Erd\H{o}s numbers are not strongly normal and provide the analogous result for Bugeaud's Mahler-inspired extension of the Copeland-Erd\H{o}s numbers. After the presentation of our results, we offer several open questions concerning normality and strong normality., Comment: 8 pages

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 only over $\mathbb{C}(z)$, but also over $\mathbb{C}(z)(\mathcal{M})$, where $\mathcal{M}$ is the set of all meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions., Comment: 15 pages, 2 figures; v3 small changes in Introduction and some fixed typos

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 $z$ goes radially to a root of unity to deduce algebraic independence results about the values of $f(z)$ at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to $F(z)$, the power series solution to the functional equation $F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0$. Specifically, we prove that the functions $F(z)$, $F(z^4)$, $F'(z)$, and $F'(z^4)$ are algebraically independent over $\mathbb{C}(z)$. An application of a celebrated result of Nishioka then allows one to replace $\mathbb{C}(z)$ by $\mathbb{Q}$ when evaluating these functions at a nonzero algebraic number $\alpha$ in the unit disc., Comment: 23 pages, 1 figure

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 \(k\)-regular sequence, we show that there is a constant \(c>0\) such that \(|f(n)|>c\log n\) infinitely often. We end our paper by answering a question of Borwein, Choi, and Coons on the sums of completely multiplicative automatic functions., Comment: 8 pages

Let $\mathcal{A}$ be a finite set of $d\times d$ matrices with integer entries and let $m_n(\mathcal{A})$ be the maximum norm of a product of $n$ elements of $\mathcal{A}$. In this paper, we classify gaps in the growth of $m_n(\mathcal{A})$; specifically, we prove that $\lim_{n\to\infty} \log m_n(\mathcal{A})/\log n\in\mathbb{Z}_{\geqslant 0}\cup\{\infty\}.$ This has applications to the growth of regular sequences as defined by Allouche and Shallit., Comment: 18 pages

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 the denominators of those rational numbers are unbounded. In particular, our result holds for the Riemann zeta function over any arithmetic progression. These results are derived via upper bounds on associated Hankel determinants., Comment: 9 pages

Suppose that $F(x)\in\mathbb{Z}[[x]]$ 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 algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If $F(x)$ is also regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$-number or a $T$-number., Comment: 52 pages

We answer a question of Calkin and Wilf concerning the maximal order of Stern's diatomic sequence. Specifically, we prove that $$\limsup_{n\to\infty}\frac{a(n)}{\varphi^{\log_2 n}}=\frac{\varphi^{\log_2 3}}{\sqrt{5}},$$ where $\varphi=(\sqrt{5}+1)/2$ is the golden ratio. This improves on previous results given by Berlekamp, Conway, and Guy, who showed that the limit value was bounded above by 1.25, and by Calkin and Wilf, who showed that the exact value was in the interval $[({\varphi}/{\sqrt{5}})({3}/{2})^{\log_2\varphi},(\varphi+1)/{\sqrt{5}}].$, Comment: 7 pages

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 the base $b$ expansion of $1/p^m$. As a consequence of this result, we prove two recent conjectures of Francisco Arag\'on, Daivd Bailey, Jonathan Borwein, and Peter Borwein concerning the base $b$ expansion of Stoneham numbers., Comment: 12 pages

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 dichotomy of Mahler functions due to Nishioka. Using our method of proof, we also provide a new proof of a Polya-Carlson type result for Mahler functions due to Rande; that is, a Mahler function which is meromorphic in the unit disk is either rational or has the unit circle as a natural boundary., Comment: 10 pages, 1 figure; added reference to Bezivin's paper

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 the special value $\C{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_n:=2^{2^n}+1$. The functions $\C{F}(z)$ and $\C{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\C{F}(\ga)$ and $\C{G}(\ga)$ are transcendental for all algebraic numbers $\ga$ which satisfy $0<\ga<1$. For a sequence $\mathbf{u}$, denote the Hankel matrix $H_n^p(\mathbf{u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$. Let $\ga$ be a real number. The {\em irrationality exponent} $\mu(\ga)$ is defined as the supremum of the set of real numbers $\mu$ such that the inequality $|\ga-p/q|

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(1)=1, and for n >= 1, s(2n)=s(n) and s(2n+1)=s(n)+s(n+1). In this note, we show that s(n) = a_1 (n) + \sum_{w in 1 (0+1)*} s([w bar]) a_{w1} (n) where w bar denotes the complement of w (obtained by sending 0 to 1 and 1 to 0, and [w] denotes the integer specified by the word w interpreted in base 2.

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 interest, we also give a way to compute the Stern value (or twisted Stern value) of a number based solely on its binary expansion., Comment: 12 pages. The new version has a corrected functional equation for $T(z)$. This changes the article significantly though the arguments remain similar

As early as the 1930s, P\'al Erd\H{o}s conjectured that: {\em for any multiplicative function $f:\mathbb{N}\to\{-1,1\}$, the partial sums $\sum_{n\leq x}f(n)$ are unbounded.} Considering this conjecture, in this paper we consider multiplicative functions $f$ satisfying $$\sum_{p\leq x}f(p)=c\cdot\frac{x}{\log x}(1+o(1)).$$ We prove that if $c>0$ then the partial sums of $f$ are unbounded, and if $c<0$ then the partial sums of $\mu f$ are unbounded. Extensions of this result are also discussed., Comment: 10 pages, newest version contains references