Clemens Müllner
YOU?
Author Swipe
View article: Synchronizing automatic sequences along Piatetski-Shapiro sequences
Synchronizing automatic sequences along Piatetski-Shapiro sequences Open
The purpose of this paper is to study subsequences of synchronizing k -automatic sequences ( a ( n )) n ≥0 along Piatetski-Shapiro Sequences ⌊ n c ⌋ with c > 1. In particular we show that ( a (⌊ n c ⌋)) n ≥0 satisfies a prime number theore…
View article: Bracket words along Hardy field sequences
Bracket words along Hardy field sequences Open
We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $ . We show that sequences thus obt…
View article: Arithmetical subword complexity of automatic sequences
Arithmetical subword complexity of automatic sequences Open
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, thes…
View article: Bracket words along Hardy field sequences
Bracket words along Hardy field sequences Open
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 seque…
View article: Gowers norms for automatic sequences
Gowers norms for automatic sequences Open
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 conne…
View article: Synchronizing automatic sequences along Piatetski-Shapiro sequences
Synchronizing automatic sequences along Piatetski-Shapiro sequences Open
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 $a(\lfloor n^c \rfloor)$ satisfi…
View article: Automaticity of uniformly recurrent substitutive sequences
Automaticity of uniformly recurrent substitutive sequences Open
We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of an underlying purely substitutive sequence. This gives an answer to a recen…
View article: Primes as sums of Fibonacci numbers
Primes as sums of Fibonacci numbers Open
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…
View article: (Logarithmic) densities for automatic sequences along primes and squares
(Logarithmic) densities for automatic sequences along primes and squares Open
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 seque…
View article: Gowers norms for automatic sequences
Gowers norms for automatic sequences Open
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 conne…
View article: Automatic sequences are orthogonal to aperiodic multiplicative functions
Automatic sequences are orthogonal to aperiodic multiplicative functions Open
Given a finite alphabet $ \mathbb{A} $ and a primitive substitution $ \theta: \mathbb{A}\to \mathbb{A}^\lambda $ (of constant length $ \lambda $), let $ (X_\theta,S) $ denote the corresponding dynamical system, where $ X_\theta $ is the cl…
View article: Automorphisms of automatic shifts
Automorphisms of automatic shifts Open
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 n…
View article: RANDOMNESS AND NON‐RANDOMNESS PROPERTIES OF PIATETSKI‐SHAPIRO SEQUENCES MODULO <i>m</i>
RANDOMNESS AND NON‐RANDOMNESS PROPERTIES OF PIATETSKI‐SHAPIRO SEQUENCES MODULO <i>m</i> Open
We study Piatetski-Shapiro sequences ([n(c)])(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 is an element of {0, 1}(k) of length k …
View article: Automatic sequences are orthogonal to aperiodic multiplicative functions
Automatic sequences are orthogonal to aperiodic multiplicative functions Open
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 $…
View article: Exponential sums with automatic sequences
Exponential sums with automatic sequences Open
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 solubi…
View article: The Rudin–Shapiro Sequence and Similar Sequences Are Normal Along Squares
The Rudin–Shapiro Sequence and Similar Sequences Are Normal Along Squares Open
We prove that digital sequences modulo m along squares are normal, which covers some prominent sequences, such as the sum of digits in base q modulo m , the Rudin–Shapiro sequence, and some generalizations. This gives, for any base, a clas…
View article: Automatic sequences fulfill the Sarnak conjecture
Automatic sequences fulfill the Sarnak conjecture Open
We present in this article a new method for dealing 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. Furt…