(Logarithmic) densities for automatic sequences along primes and squares

Adamczewski, Boris; Drmota, Michael; Müllner, Clemens

doi:10.1090/tran/8476

(Logarithmic) densities for automatic sequences along primes and squares
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by Boris Adamczewski, Michael Drmota and Clemens Müllner PDF

Trans. Amer. Math. Soc. 375 (2022), 455-499 Request permission

Abstract:

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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