The Sum of Digits Function in Number Fields: Distribution in Residue Classes

Abstract

Canonical number systems are the natural generalization ofq-adic number systems to number fields. Such number systems admit a certain representation of each algebraic integer of a given number field with respect to the powers of a given base numberb. The aim of this paper is to study the sum of digits functionνb(γ) of these number systems. In particular, we prove, thatνb(γ) is equidistributed in residue classes modulomin any ideal s. Furthermore, we investigate the sum of digits function over sumsets.