{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/381050"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/381050","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Arithmetic regularity lemmas and applications","abstract":"This thesis investigates various aspects of arithmetic regularity lemmas in the context of vector spaces over finite fields of prime characteristic. Chapter 1 obtains a generalisation of the induced arithmetic removal lemma of Bhattacharyya, Fischer, and Lovett [6] for translation-invariant arithmetic patterns, extending it to partition-regular patterns of complexity 1; this also strengthens the result of Fox, Tidor and Zhao [16] for general complexity-1 patterns. Chapter 2 establishes a wowzer-type lower bound on the size of the partition arising from the so-called strong arithmetic regularity lemma, which matches the bound of Conlon and Fox [12] for the analogous result in the graph-theoretic setting. The rest of the thesis concerns higher-order arithmetic regularity, in particular undertaking a study of local higher-order uniformity in Chapter 3. Two approaches to defining local uniformity on polynomial factors are proposed and subsequently applied to generalise two theorems of Green and Sanders [28] to polynomial factors of all degrees; specifically, it is shown that given any bounded function on a vector space over a field of characteristic 2, there is always a polynomial factor on whose zero atom the function is uniform, while over fields of characteristic greater than 2 this cannot be guaranteed. Finally, Chapters 4 and 5 address several questions concerning the quadratic arithmetic regularity lemmas of Terry and Wolf [60] under model-theoretically motivated tameness assumptions, including a proof of their conjecture that the set referred to as the quadratic Green-Sanders example has bounded VC₂-dimension.","abstract_html":"This thesis investigates various aspects of arithmetic regularity lemmas in the context of vector spaces over finite fields of prime characteristic. Chapter 1 obtains a generalisation of the induced arithmetic removal lemma of Bhattacharyya, Fischer, and Lovett [6] for translation-invariant arithmetic patterns, extending it to partition-regular patterns of complexity 1; this also strengthens the result of Fox, Tidor and Zhao [16] for general complexity-1 patterns. Chapter 2 establishes a wowzer-type lower bound on the size of the partition arising from the so-called strong arithmetic regularity lemma, which matches the bound of Conlon and Fox [12] for the analogous result in the graph-theoretic setting. The rest of the thesis concerns higher-order arithmetic regularity, in particular undertaking a study of local higher-order uniformity in Chapter 3. Two approaches to defining local uniformity on polynomial factors are proposed and subsequently applied to generalise two theorems of Green and Sanders [28] to polynomial factors of all degrees; specifically, it is shown that given any bounded function on a vector space over a field of characteristic 2, there is always a polynomial factor on whose zero atom the function is uniform, while over fields of characteristic greater than 2 this cannot be guaranteed. Finally, Chapters 4 and 5 address several questions concerning the quadratic arithmetic regularity lemmas of Terry and Wolf [60] under model-theoretically motivated tameness assumptions, including a proof of their conjecture that the set referred to as the quadratic Green-Sanders example has bounded VC₂-dimension.","abstract_has_math":false,"creators":["Gladkova, Valeriia"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Wolf, Julia"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-08-31","date_published":"2024-08-31","updated_at":"2026-07-22T22:24:10Z","subjects":["Additive combinatorics","Combinatorics","Higher-order Fourier Analysis","Regularity Lemmas"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/19e336c7-c441-44b8-8aa9-547330cdb484/download","http://purl.org/NET/rdflicense/allrightsreserved"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.116416","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wolf, Julia"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Harding Distinguished Postgraduate Scholars Programme"]},{"key":"dc:creator","label":"Author","values":["Gladkova, Valeriia"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2024-08-31"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/381050"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Additive combinatorics","Combinatorics","Higher-order Fourier Analysis","Regularity Lemmas"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/19e336c7-c441-44b8-8aa9-547330cdb484/download","http://purl.org/NET/rdflicense/allrightsreserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.116416"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/889867eb-2a97-4171-aa97-25978650b731/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis investigates various aspects of arithmetic regularity lemmas in the context of vector spaces over finite fields of prime characteristic. Chapter 1 obtains a generalisation of the induced arithmetic removal lemma of Bhattacharyya, Fischer, and Lovett [6] for translation-invariant arithmetic patterns, extending it to partition-regular patterns of complexity 1; this also strengthens the result of Fox, Tidor and Zhao [16] for general complexity-1 patterns. Chapter 2 establishes a wowzer-type lower bound on the size of the partition arising from the so-called strong arithmetic regularity lemma, which matches the bound of Conlon and Fox [12] for the analogous result in the graph-theoretic setting. The rest of the thesis concerns higher-order arithmetic regularity, in particular undertaking a study of local higher-order uniformity in Chapter 3. Two approaches to defining local uniformity on polynomial factors are proposed and subsequently applied to generalise two theorems of Green and Sanders [28] to polynomial factors of all degrees; specifically, it is shown that given any bounded function on a vector space over a field of characteristic 2, there is always a polynomial factor on whose zero atom the function is uniform, while over fields of characteristic greater than 2 this cannot be guaranteed. Finally, Chapters 4 and 5 address several questions concerning the quadratic arithmetic regularity lemmas of Terry and Wolf [60] under model-theoretically motivated tameness assumptions, including a proof of their conjecture that the set referred to as the quadratic Green-Sanders example has bounded VC₂-dimension."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["db5b11e84fbae6508ce5d946ee620a9e","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Arithmetic regularity lemmas and applications"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wolf, Julia"],"dc:contributor.sponsor":["Harding Distinguished Postgraduate Scholars Programme"],"dc:creator":["Gladkova, Valeriia"],"dc:date.issued":["2024-08-31"],"dc:description.abstract":["This thesis investigates various aspects of arithmetic regularity lemmas in the context of vector spaces over finite fields of prime characteristic. Chapter 1 obtains a generalisation of the induced arithmetic removal lemma of Bhattacharyya, Fischer, and Lovett [6] for translation-invariant arithmetic patterns, extending it to partition-regular patterns of complexity 1; this also strengthens the result of Fox, Tidor and Zhao [16] for general complexity-1 patterns. Chapter 2 establishes a wowzer-type lower bound on the size of the partition arising from the so-called strong arithmetic regularity lemma, which matches the bound of Conlon and Fox [12] for the analogous result in the graph-theoretic setting. The rest of the thesis concerns higher-order arithmetic regularity, in particular undertaking a study of local higher-order uniformity in Chapter 3. Two approaches to defining local uniformity on polynomial factors are proposed and subsequently applied to generalise two theorems of Green and Sanders [28] to polynomial factors of all degrees; specifically, it is shown that given any bounded function on a vector space over a field of characteristic 2, there is always a polynomial factor on whose zero atom the function is uniform, while over fields of characteristic greater than 2 this cannot be guaranteed. Finally, Chapters 4 and 5 address several questions concerning the quadratic arithmetic regularity lemmas of Terry and Wolf [60] under model-theoretically motivated tameness assumptions, including a proof of their conjecture that the set referred to as the quadratic Green-Sanders example has bounded VC₂-dimension."],"dc:format.checksum.md5":["db5b11e84fbae6508ce5d946ee620a9e","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.116416"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/889867eb-2a97-4171-aa97-25978650b731/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/381050"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/19e336c7-c441-44b8-8aa9-547330cdb484/download","http://purl.org/NET/rdflicense/allrightsreserved"],"dc:subject":["Additive combinatorics","Combinatorics","Higher-order Fourier Analysis","Regularity Lemmas"],"dc:title":["Arithmetic regularity lemmas and applications"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:10Z"}