This diploma thesis discusses a problem related to the well-known third Gelfond problem. We consider the generalized Thue-Morse sequence, i.e. the sum-of-digits function to the base q modulo m, and show that each subsequence along squares of length k appears with asymptotic frequency 1/q k.
The first chapter gives some general information about the sum-of-digits function as well as the Gelfond problems. Furthermore, an outline of the complete proof and a more detailed description of the following chapters are covered.
We use a method developed by Mauduit and Rivat in 2009 which involves Fourier-analytic methods. A similar problem, i.e. the special case q=m=2, was already solved by Drmota, Mauduit and Rivat in 2014.
The main contribution of this work is to find appropriate bounds for the corresponding Fourier terms in this more general setting. This is covered in Chapter 2. Whereas the proof of Drmota, Mauduit and Rivat, for the special case, consists of finding one special sequence, we have to consider many such admissible sequences for our generalization.
In Chapter 3, we discuss some auxiliary results and, finally, in Chapter 4, we cover the proof of the result stated above. This part is quite similar to the proof of the special case q=m=2. We conclude this work by suggesting various possible generalizations.