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 autocorrelations, and showed that the average value of the autocorrelations is zero. In particular, they showed that & sum;(n <= x)|eta(n)|=o(x(alpha)) for any alpha> log (3)/ log (4). In this paper, we sharpen this result, providing upper and lower bounds for alpha. On the way to our lower bounds, we obtain the structure of the linear representation of the point-wise product of two k-regular sequences, which may be of independent interest.

We give a new proof of the singular continuity of Minkowski’s $?$ -function. Our proof follows by showing that the maximal Lyapunov exponent of a specific pair of $3\times 3$ nonnegative integer matrices related to Stern’s diatomic sequence is strictly greater than $\log 2$ .

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.

The Thue-Morse sequence {t(n)}(n >= 0) is the indicator function of the parity of the number of ones in the binary expansion of nonnegative 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 > root phi = 1.272019 ..., the set of the numbers 1, Sigma(n >= 1) t(n)/beta(n), Sigma(n >= 1)t(n(2))/beta(n), ..., Sigma(n >= 1)t(n(k))/beta(n), ... is linearly independent over the field Q(beta), where phi := (1 + root 5)/2 is the golden ratio. Our result yields that for any integer k >= 1 and for any a(1), a(2),..., a(k) is an element of Q(beta), not all zero, the sequence {a(1)t(n) + a(2)t(n(2)) + ... + a(k)t(n(k))}(n >= 1) cannot be eventually periodic.

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.

We give a new proof of a theorem of Bell and Coons ['Transcendence tests for Mahler functions', Proc. Amer. Math. Soc. 145(3) (2017), 1061-1070] on the leading order radial asymptotics of Mahler functions that are the generating functions of regular sequences. Our method allows us to provide a description of the oscillations whose existence was shown by Bell and Coons. This extends very recent results of Poulet and Rivoal ['Radial behavior of Mahler functions', Int. J. Number Theory, to appear].

Motivated by near-identical graphs of two increasing continuous functions-one related to Zaremba's conjecture and the other due to Salem-we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem's strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.

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.

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.

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.

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 Mañibo. 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.

For some time now, I have been trying to understand the complexity of integer sequences from a variety of different viewpoints and, at least at some level, trying to reconcile these viewpoints. However vague that sounds—and it certainly is vague to me—in this short note, I hope to explain this sentiment.

We show that a missing q-ary digit set F subset of [0, 1] has a 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.

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.

Aperiodic order is a relatively young area of mathematics with connections to many other fields, including discrete geometry, harmonic analysis, dynamical systems, algebra, combinatorics and, above all, number theory. In fact, numbertheoretic methods and results are present in practically all of these connections. It was one aim of this workshop to review, strengthen and foster these connections.

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

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.

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.