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Clemens Müllner

Publications and source records attributed to Clemens Müllner.

18 recordsLinked to original sources
arXiv2026

Automaticity of uniformly recurrent substitutive sequences

We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question posed by Allouche, Dekking and Queffélec in the uniformly recurrent case. We show that the same criterion characterizes automaticity of minimal substitutive systems. Furthermore, we construct a minimal substitutive system whose maximal equicontinuous factor is the 2-adic odometer, and for which the corresponding factor map is everywhere uncountable-to-one. We conjecture that a minimal substitutive system is k-automatic if and only if it is an everywhere finite-to-one extension of a k-adic odometer.

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arXiv2024

Arithmetical subword complexity of automatic sequences

We fully classify automatic sequences $a$ over a finite alphabet $Ω$ with the property that each word over $Ω$ appears is $a$ along an arithmetic progression. Using the terminology introduced by Avgustinovich, Fon-Der-Flaass and Frid, these are the automatic sequences with the maximal possible arithmetical subword complexity. More generally, we obtain an asymptotic formula for arithmetical (and even polynomial) subword complexity of a given automatic sequence $a$.

math.NT↗
arXiv2023

Gowers norms for automatic sequences

We show that any automatic sequence can be separated into a structured part and a Gowers uniform part in a way that is considerably more efficient than guaranteed by the Arithmetic Regularity Lemma. For sequences produced by strongly connected and prolongable automata, the structured part is rationally almost periodic, while for general sequences the description is marginally more complicated. In particular, we show that all automatic sequences orthogonal to periodic sequences are Gowers uniform. As an application, we obtain for any $l \geq 2$ and any automatic set $A \subset \mathbb{N}_0$ lower bounds on the number of $l$-term arithmetic progressions - contained in $A$ - with a given difference. The analogous result is false for general subsets of $\mathbb{N}_0$ and progressions of length $\geq 5$.

math.NT↗
arXiv2023

Bracket words along Hardy field sequences

We study bracket words, which are a far-reaching generalisation of Sturmian words, along Hardy field sequences, which are a far-reaching generalisation of Piatetski--Shapiro sequences $\lfloor n^c \rfloor$. We show that thus obtained sequences are deterministic (i.e., they have sub-exponential subword complexity) and satisfy Sarnak's conjecture.

math.NT↗
arXiv2022

Synchronizing automatic sequences along Piatetski-Shapiro sequences

The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c \rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c \rfloor)$ satisfies a prime number theorem of the form $\sum_{n\le x} Λ(n)a(\lfloor n^c \rfloor) \sim C\, x$, and, furthermore, that it is deterministic for $c \in \mathbb R\setminus \mathbb Z$. As an interesting additional result, we show that the sequence $\lfloor n^c\rfloor \bmod m$ has polynomial subword complexity.

math.NT↗
arXiv2022

Primes as sums of Fibonacci numbers

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as the sum of $k$ different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. The proof of such a prime number theorem, combined with a corresponding local result, is the central contribution of this paper, from which we derive the result stated in the beginning. Problems of this type have been discussed intensively in the context of the base-$q$ expansion. The Gelfond problems (1968/1969), and the Sarnak conjecture, were the driving forces of this development. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009), thus leaving open only part of the third Gelfond problem. Later the second author (2017) proved Sarnak's conjecture for all automatic sequences, which are based on the q-ary expansion of integers, and which generalize the sum-of-digits function in base $q$ considerably. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat's method considerably. In fact, we are departing significantly from this method, proving the statement that $\exp(2πi \vartheta\mathsf z(n))$ has \emph{level of distribution} $1$ (here $\mathsf z(n)$ is the number of Fibonacci numbers needed to write $n$ as their sum). This latter result forms an essential part of our treatment of the occurring sums of type $\textrm I$ and $\textrm{II}$ and uses Gowers norms related to $\mathsf z(n)$ as a central technical tool. The appearance of Gowers norms in our method is intimately tied to the iterated application of a new generalization of van der Corput's inequality.

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arXiv2021

(Logarithmic) densities for automatic sequences along primes and squares

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lemańczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function.

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arXiv2020

The VC-Dimension of Axis-Parallel Boxes on the Torus

We show in this paper that the VC-dimension of the family of $d$-dimensional axis-parallel boxes and cubes on the $d$-dimensional torus are both asymptotically $d \log_2(d)$. This is especially surprising as the VC-dimension usually grows linearly with $d$ in similar settings.

math.CO↗
arXiv2019

Automorphisms of automatic shifts

In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from {\em twisted} compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a letter-to-letter code.

math.DS↗
arXiv2019

Palindromic Subsequences in Finite Words

In 1999 Lyngsø and Pedersen proposed a conjecture stating that every binary circular word of length $n$ with equal number of zeros and ones has an antipalindromic linear subsequence of length at least $\frac{2}{3}n$. No progress over a trivial $\frac{1}{2}n$ bound has been achieved since then. We suggest a palindromic counterpart to this conjecture and provide a non-trivial infinite series of circular words which prove the upper bound of $\frac{2}{3}n$ for both conjectures at the same time. The construction also works for words over an alphabet of size $k$ and gives rise to a generalization of the conjecture by Lyngsø and Pedersen. Moreover, we discuss some possible strengthenings and weakenings of the named conjectures. We also propose two similar conjectures for linear words and provide some evidences for them.

cs.FL↗
arXiv2018

Randomness and non-randomness properties of Piatetski-Shapiro sequences modulo m

We study Piatetski-Shapiro sequences $(\lfloor n^c\rfloor)_n$ modulo m, for non-integer $c >1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in\{0,1\}^k$ of length $k < c + 1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi-Kátai criterion, we prove that the sequence $\lfloor n^c\rfloor$ modulo m is asymptotically orthogonal to multiplicative functions bounded by $1$ and with mean value $0$.

math.NT↗
arXiv2018

Automatic sequences are orthogonal to aperiodic multiplicative functions

Given a finite alphabet $\mathbb{A}$ and a primitive substitution $θ:\mathbb{A}\to\mathbb{A}^λ$ (of constant length $λ$), let $(X_θ,S)$ denote the corresponding dynamical system, where $X_θ$ is the closure of the orbit via the left shift $S$ of a fixed point of the natural extension of $θ$ to a self-map of $\mathbb{A}^{\mathbb{Z}}$. The main result of the paper is that all continuous observables in $X_θ$ are orthogonal to any bounded, aperiodic, multiplicative function $\mathbf{u}:\mathbb{N}\to\mathbb{C}$, i.e. \[ \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(S^nx)\mathbf{u}(n)=0\] for all $f\in C(X_θ)$ and $x\in X_θ$. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.

math.DS↗
arXiv2017

Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences, II

We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $ω\in\{0,1\}^L$, where $L\geq 1$, the set of occurrences of $ω$ as a subword (contiguous finite subsequence) of the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ has asymptotic density $2^{-L}$. This is an improvement over a recent result by the second author, which handles the case $1<c<4/3$. In particular, this result shows that for $1<c<3/2$ the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ attains both of its values with asymptotic density $1/2$, which improves on the bound $c<1.4$ obtained by Mauduit and Rivat (who obtained this bound in the more general setting of $q$-multiplicative functions, however) and on the bound $c\leq 1.42$ obtained by the second author. In the course of proving the main theorem, we show that $2/3$ is an admissible level of distribution for the Thue--Morse sequence, that is, it satisfies a Bombieri--Vinogradov type theorem for each exponent $η<2/3$. This improves on a result by Fouvry and Mauduit, who obtained the exponent $0.5924$. Moreover, the underlying theorem implies that every finite word $ω\in\{0,1\}^L$ is contained as an arithmetic subsequence of $\mathbf t$.

math.NT↗
arXiv2017

The Rudin-Shapiro sequence and similar sequences are normal along squares

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

math.NT↗
arXiv2017

Exponential sums with automatic sequences

We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type $e_q(f(n))$, where $f$ is a rational fraction, in the Pólya-Vinogradov range. This applies to Kloosterman sums, and may be used to study solubility of congruence equations over automatic sequences. We obtain this as consequence of a general result, stating that sums over automatic sequences can be bounded effectively in terms of two-point correlation sums over intervals.

math.NT↗
arXiv2017

Automatic sequences fulfill the Sarnak conjecture

We present in this paper a new method to deal with automatic sequences. This method allows us to prove a Möbius-randomness-principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a Prime Number Theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to $0$.

math.NT↗
arXiv2017

Möbius orthogonality for the Zeckendorf sum-of-digits function

We show that the (morphic) sequence $(-1)^{s_φ(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_φ$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n<N} (-1)^{s_φ(n)} μ(n) = o(N)$, that is, this sequence satisfies the Sarnak conjecture.

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