The sum-of-digits function of canonical number systems: Distribution in residue classes

In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sárközy to this kind of numeration systems.