(Logarithmic) densities for automatic sequences along primes and squares

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 $(n^2{)}_{n\ge 0}$ and primes $(p_n{)}_{n\ge 1}$ 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.

35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences