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<p>The purpose of this paper is to study subsequences of synchronizing <i>k</i>-automatic sequences (<i>a</i>(<i>n</i>))<sub><i>n</i>≥0</sub> along Piatetski-Shapiro Sequences ⌊<i>n</i><sup><i>c</i></sup>⌋ with <i>c</i> &gt; 1. In particular we show that (<i>a</i>(⌊<i>n</i><sup><i>c</i></sup>⌋))<sub><i>n</i>≥0</sub> satisfies a prime number theorem of the form ∑<sub><i>n</i>≤<i>x</i></sub> Λ(<i>n</i>)<i>a</i>(⌊<i>n</i><sup><i>c</i></sup>⌋) ∼ <i>Cx</i>, and, furthermore, that it is deterministic. As an interesting additional result we show that the sequence (⌊<i>n</i><sup><i>c</i></sup>⌋ mod <i>m</i>)<sub><i>n</i>≥0</sub> has polynomial subword complexity.</p>

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Synchronizing automatic sequences along Piatetski-Shapiro sequences

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  • 发表日期：

    2025年6月29日

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  • Jean-Marc Deshouillers,
  • Michael Drmota,
  • Clemens Müllner,
  • Andrei Shubin,
  • Lukas Spiegelhofer

摘要

The purpose of this paper is to study subsequences of synchronizing k-automatic sequences (a(n))n≥0 along Piatetski-Shapiro Sequences ⌊nc⌋ with c > 1. In particular we show that (a(⌊nc⌋))n≥0 satisfies a prime number theorem of the form ∑n≤x Λ(n)a(⌊nc⌋) ∼ Cx, and, furthermore, that it is deterministic. As an interesting additional result we show that the sequence (⌊nc⌋ mod m)n≥0 has polynomial subword complexity.