Abstract:
It is well known that the every letter $\alpha$ of an automatic
sequence $a(n)$ has a logarithmic density -- and it can be decided
when this logarithmic density is actually a density. For example, the
letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both
frequencies $1/2$.

The purpose of this talk is to present a corresponding result for
subsequences of general automatic sequences along primes and squares.
This is a far reaching generalization of two breakthrough results of
Mauduit and Rivat from 2009 and 2010, where they solved two conjectures
by Gelfond on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^{^2})$
(where $p_n$ denotes the sequence of primes).

More technically, one has to develop a method to transfer density
results for primitive automatic sequences to logarithmic-density
results for general automatic sequences. Then, as an application,
one can deduce 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, if densities exist then they are (usually) rational.

This is joint work with Boris Adamczewski  and Clemens Müllner.

Zoom-Meeting beitreten:
https://zoom.us/j/93778343609?pwd=YngwNmNLZWgxczRnYXNxdHZTSVdSdz09

Meeting-ID: 937 7834 3609
Kenncode: BCfC2J