(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≥0 and primes (p n ) n≥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.