{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/116173"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/116173","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Pila-Wilkie theorem and analytic Ax-Kochen-Ersov theory","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled &#x27;Open Access&#x27;. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_has_math":false,"creators":["Bhardwaj, Neer"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["van den Dries, Lou","Hieronymi, Philipp","Heller, Jeremiah","Freitag, James"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-08","date_published":"2022-08","updated_at":"2026-07-22T22:24:55Z","subjects":["Logic","Model Theory","Square-free integers","O-minimality","Pila-Wilkie counting theorem","Valued fields with analytic structure","Ax-Kochen-Ersov principle"],"languages":["en","eng"],"rights":["Copyright 2022 Neer Bhardwaj"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/116173","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["van den Dries, Lou","Hieronymi, Philipp","Heller, Jeremiah","Freitag, James"]},{"key":"dc:creator","label":"Author","values":["Bhardwaj, Neer"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-08","2022-06-30"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Logic","Model Theory","Square-free integers","O-minimality","Pila-Wilkie counting theorem","Valued fields with analytic structure","Ax-Kochen-Ersov principle"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2022 Neer Bhardwaj"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/116173"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Neer Bhardwaj, accepted the attached license on 2022-06-27 at 16:16.","The student, Neer Bhardwaj, submitted this Dissertation for approval on 2022-06-27 at 16:33.","This Dissertation was approved for publication on 2022-06-30 at 15:31.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18112 on 2022-11-15 at 17:37:50","In Part I, we consider the four structures $(\\mathbb{Z}; \\SF)$, $(\\mathbb{Z}; <, \\SF)$, $(\\mathbb{Q}; \\SQ)$, and $(\\mathbb{Q}; <, \\SQ)$ where $\\mathbb{Z}$ is the additive group of integers, $\\SF$ is the set of $a \\in \\mathbb{Z}$ such that $v_{p}(a) < 2$ for every prime $p$ and corresponding $p$-adic valuation $v_{p}$, $\\mathbb{Q}$ and $\\SQ$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. Part II gives an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov. Part III develops an extension theory for analytic valuation rings in order to establish Ax-Kochen-Ersov type results for these structures. New is that we can add in salient cases lifts of the residue field and the value group and show that the induced structure on the lifted residue field is just its field structure, and on the lifted value group is just its ordered abelian group structure. This restores an analogy with the non-analytic AKE-setting that was missing in earlier treatments of analytic AKE-theory."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The Pila-Wilkie theorem and analytic Ax-Kochen-Ersov theory"]}]}],"canonical_facts":{"dc:contributor":["van den Dries, Lou","Hieronymi, Philipp","Heller, Jeremiah","Freitag, James"],"dc:creator":["Bhardwaj, Neer"],"dc:date":["2022-08","2022-06-30"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Neer Bhardwaj, accepted the attached license on 2022-06-27 at 16:16.","The student, Neer Bhardwaj, submitted this Dissertation for approval on 2022-06-27 at 16:33.","This Dissertation was approved for publication on 2022-06-30 at 15:31.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18112 on 2022-11-15 at 17:37:50","In Part I, we consider the four structures $(\\mathbb{Z}; \\SF)$, $(\\mathbb{Z}; <, \\SF)$, $(\\mathbb{Q}; \\SQ)$, and $(\\mathbb{Q}; <, \\SQ)$ where $\\mathbb{Z}$ is the additive group of integers, $\\SF$ is the set of $a \\in \\mathbb{Z}$ such that $v_{p}(a) < 2$ for every prime $p$ and corresponding $p$-adic valuation $v_{p}$, $\\mathbb{Q}$ and $\\SQ$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. Part II gives an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov. Part III develops an extension theory for analytic valuation rings in order to establish Ax-Kochen-Ersov type results for these structures. New is that we can add in salient cases lifts of the residue field and the value group and show that the induced structure on the lifted residue field is just its field structure, and on the lifted value group is just its ordered abelian group structure. This restores an analogy with the non-analytic AKE-setting that was missing in earlier treatments of analytic AKE-theory."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/116173"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Neer Bhardwaj"],"dc:subject":["Logic","Model Theory","Square-free integers","O-minimality","Pila-Wilkie counting theorem","Valued fields with analytic structure","Ax-Kochen-Ersov principle"],"dc:title":["The Pila-Wilkie theorem and analytic Ax-Kochen-Ersov theory"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:55Z"}