(Logarithmic) densities for automatic sequences along primes and squares
| Autor: | Boris Adamczewski, Clemens Müllner, Michael Drmota |
|---|---|
| Přispěvatelé: | Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Combinatoire, théorie des nombres (CTN), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), European Project: 648132,H2020,ERC-2014-CoG,ANT(2015) |
| Rok vydání: | 2021 |
| Předmět: |
FOS: Computer and information sciences
Automatic sequence Mathematics - Number Theory Logarithm Formal Languages and Automata Theory (cs.FL) Applied Mathematics General Mathematics 010102 general mathematics Multiplicative function Computer Science - Formal Languages and Automata Theory Function (mathematics) 01 natural sciences [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] Prime (order theory) Combinatorics Transfer (group theory) Aperiodic graph Bounded function Primary: 11B85 11L20 11N05 Secondary: 11A63 11L03 FOS: Mathematics Number Theory (math.NT) 0101 mathematics Mathematics |
| Zdroj: | Transactions of the American Mathematical Society Transactions of the American Mathematical Society, 2021, pp.1. ⟨10.1090/tran/8476⟩ |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/tran/8476 |
| Popis: | 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 Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function. Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences |
| Databáze: | OpenAIRE |
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