{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86975"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86975","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Distribution of Generalized Sum -of -Digits Functions","abstract":"\"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.\"","abstract_html":"&quot;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 &quot;&quot;base- Q&quot;&quot; representation of the form n=j≥0aj nQj with &quot;&quot;digits&quot;&quot; aj( n) satisfying 0≤ajn&lt;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,&amp;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.&quot;","abstract_has_math":false,"creators":["Hoit, Abigail"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hildebrand, A.J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:26Z","date_published":"2015-09-28T15:20:26Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9944880"],"render_values":[{"text":"(MiAaPQ)AAI9944880","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86975","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hildebrand, A.J."]},{"key":"dc:creator","label":"Author","values":["Hoit, Abigail"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:26Z","10000-01-01","1999"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86975","(MiAaPQ)AAI9944880"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"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.\"","Made available in DSpace on 2015-09-28T15:20:26Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9944880.pdf: 2640909 bytes, checksum: a5a9dc5eda8485c484b31ad425f07ffe (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 88256 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","62 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."]},{"key":"dc:title","label":"Title","values":["The Distribution of Generalized Sum -of -Digits Functions"]}]}],"canonical_facts":{"dc:contributor":["Hildebrand, A.J."],"dc:creator":["Hoit, Abigail"],"dc:date":["2015-09-28T15:20:26Z","10000-01-01","1999"],"dc:description":["\"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.\"","Made available in DSpace on 2015-09-28T15:20:26Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9944880.pdf: 2640909 bytes, checksum: a5a9dc5eda8485c484b31ad425f07ffe (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 88256 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","62 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."],"dc:identifier":["http://hdl.handle.net/2142/86975","(MiAaPQ)AAI9944880"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["The Distribution of Generalized Sum -of -Digits Functions"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:28Z"}