Additive functions for number systems in function fields

Abstract

Let Fq be a finite field with q elements and p ∈ Fq[X, Y ]. In this paper we study properties of additive functions with respect to number systems which are defined in the ring Fq[X, Y ]/p Fq[X, Y ]. Our results comprise distribution results, exponential sum estimations as well as a version of Waring's Problem restricted by such additive functions. Similar results have been shown for b-adic number systems as well as number systems in finite fields in the sense of Kovács and Pethő. In the proofs of the results contained in the present paper new difficulties occur because the "fundamental domains" associated to the number systems studied here have a complicated structure.

On a second conjecture of Stolarsky: the sum of digits of polynomial values

“…These conditions are automatically verified by the sum-of-digits function. Furthermore we remark that by replacing the theorem of Bassily and Kátai (Theorem 2.2) by analogous results due to Drmota and Steiner [4], Madritsch [7], Madritsch and Pethő [8], Drmota and Gutenbrunner [3], and Madritsch and Thuswaldner [10], respectively, one can prove analogous results for additive functions of polynomials in numeration systems that are defined via linear recurrent sequences (such as the Zeckendorf expansion), for additive functions of polynomials in numeration systems in algebraic number fields, for additive functions of polynomials in numeration systems in the quotient ring of polynomials over Z, for additive functions of polynomials in numeration systems in the ring of polynomials over a finite field and for additive functions of polynomials in numeration systems in function fields, respectively.…”