15 research outputs found
Arithmetical subword complexity of automatic sequences
We fully classify automatic sequences over a finite alphabet
with the property that each word over appears is along an
arithmetic progression. Using the terminology introduced by Avgustinovich,
Fon-Der-Flaass and Frid, these are the automatic sequences with the maximal
possible arithmetical subword complexity. More generally, we obtain an
asymptotic formula for arithmetical (and even polynomial) subword complexity of
a given automatic sequence .Comment: 14 pages, comments welcom
Synchronizing automatic sequences along Piatetski-Shapiro sequences
The purpose of this paper is to study subsequences of synchronizing
-automatic sequences along Piatetski-Shapiro sequences with non-integer . In particular, we show that satisfies a prime number theorem of the form , and, furthermore, that it is
deterministic for . As an interesting
additional result, we show that the sequence has
polynomial subword complexity.Comment: 32 page
(Logarithmic) densities for automatic sequences along primes and squares
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 and primes
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: 38 page