The Distribution of Generalized Sum -of -Digits Functions

"Let Q=Qjinfinity j=0 be a strictly increasing sequence of integers with Q 0 = 1 and such that each Qj is a divisor of Qj+1 . The sequence Q is a numeration system in the sense that every positive integer n has a unique ""base- Q"" representation of the form n=j≥0aj nQj with ""digits"" aj( n) satisfying 0≤ajn<Q j+1/Qj . A Q-additive function is a function f:N→ C of the form fn=j≥ 0fjaj n where n=j≥0aj nQj is the base-Q representation of n and the component functions fj are defined on 0,1,&ldots;, Qj+1/Qj-1 and satisfy fj(0) = 0. These functions generalize the sum-of-digits functions for base-Q representations, defined by sQn= j≥0ajn . We consider the distribution of integer-valued Q-additive functions in residue classes modulo a prime m and also the distribution of real-valued Q-additive functions modulo 1. In each case, we give necessary and sufficient conditions for a Q -additive function to be uniformly (resp. non-uniformly) distributed. We also introduce so-called block product sequences, which are sequences obtained by a certain algebraic operation on finite blocks of digits, and show that these sequences are closely related to Q-additive functions."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Hoit, Abigail

Contributors dc:contributor

• Hildebrand, A.J.

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI9944880

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86975