{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120122"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120122","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Two topics in arithmetic combinatorics","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","abstract_has_math":false,"creators":["Roy, Souktik"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Balogh, Jozsef","Kostochka, Alexandr","Ford, Kevin","Bradshaw, Peter"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:56Z","subjects":["Sum Product","O-minimality","Sidon Sets"],"languages":["en","eng"],"rights":["Copyright 2023 Souktik Roy"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120122","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Balogh, Jozsef","Kostochka, Alexandr","Ford, Kevin","Bradshaw, Peter"]},{"key":"dc:creator","label":"Author","values":["Roy, Souktik"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2023-04-28"]},{"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":["Sum Product","O-minimality","Sidon Sets"]}]},{"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 2023 Souktik Roy"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120122"]}]},{"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 2023-09-01 without embargo terms","The student, Souktik Roy, accepted the attached license on 2023-04-27 at 10:47.","The student, Souktik Roy, submitted this Dissertation for approval on 2023-04-27 at 11:06.","This Dissertation was approved for publication on 2023-04-28 at 11:27.","DSpace SAF Submission Ingestion Package generated from Vireo submission #19208 on 2023-09-01 at 16:55:41","This thesis - naturally delineated into two parts - attempts to address two flavors of problems in arithmetic combinatorics. These two parts are based on the papers \\cite{JRT} and \\cite{BFR}, respectively - all mathematical content in this thesis has already appeared in these papers (and associated preprints). In the first, together with Jing and Tran \\cite{JRT} we find inspiration in the sum-product phenomenon and the classification of two-variable polynomials of bounded growth. For a bivariate $P(x,y) \\in \\RR[x,y]\\setminus (\\RR[x] \\cup \\RR[y])$, our first result shows that for all finite $A \\subseteq \\RR$, $|P(A,A)|\\geq \\alpha|A|^{5/4}$ with $\\alpha =\\alpha(\\deg P) \\in \\RR^{>0}$ unless $$ P(x,y)=f(\\gamma u(x)+\\delta u(y)) \\text{ or } P(x,y)=f(u^m(x)u^n(y)) $$ for some univariate $f, u \\in \\RR[t]\\setminus \\RR$, constants $\\gamma, \\delta \\in \\RR^{\\neq 0}$, and $m, n\\in \\NN^{\\geq 1}$. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results in this chapter are sum-product type theorems for two polynomials, generalizing the classical result by Erd\\H os and Szemer\\'edi as well as a theorem by Shen. We also obtain similar results for $\\CC$, and from this deduce results for fields of characteristic $0$ and fields of large prime characteristic. We use tools from semialgebraic/o-minimal geometry to prove these results; exposition is provided on these methods to make them accessible to readers primarily concerned with combinatorics. In the second, together with Balogh and F\\\"uredi \\cite{BFR}, we combine two elementary proofs to show that the maximum size of a Sidon set of $\\{ 1, 2, \\ldots, n\\}$ is at most $\\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$ - the first non-constant improvement of the error term $n^{1/4}$ in this classical combinatorial number theory problem since 1969. This implies improvements in some related problems which are also discussed."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Two topics in arithmetic combinatorics"]}]}],"canonical_facts":{"dc:contributor":["Balogh, Jozsef","Kostochka, Alexandr","Ford, Kevin","Bradshaw, Peter"],"dc:creator":["Roy, Souktik"],"dc:date":["2023-05","2023-04-28"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","The student, Souktik Roy, accepted the attached license on 2023-04-27 at 10:47.","The student, Souktik Roy, submitted this Dissertation for approval on 2023-04-27 at 11:06.","This Dissertation was approved for publication on 2023-04-28 at 11:27.","DSpace SAF Submission Ingestion Package generated from Vireo submission #19208 on 2023-09-01 at 16:55:41","This thesis - naturally delineated into two parts - attempts to address two flavors of problems in arithmetic combinatorics. These two parts are based on the papers \\cite{JRT} and \\cite{BFR}, respectively - all mathematical content in this thesis has already appeared in these papers (and associated preprints). In the first, together with Jing and Tran \\cite{JRT} we find inspiration in the sum-product phenomenon and the classification of two-variable polynomials of bounded growth. For a bivariate $P(x,y) \\in \\RR[x,y]\\setminus (\\RR[x] \\cup \\RR[y])$, our first result shows that for all finite $A \\subseteq \\RR$, $|P(A,A)|\\geq \\alpha|A|^{5/4}$ with $\\alpha =\\alpha(\\deg P) \\in \\RR^{>0}$ unless $$ P(x,y)=f(\\gamma u(x)+\\delta u(y)) \\text{ or } P(x,y)=f(u^m(x)u^n(y)) $$ for some univariate $f, u \\in \\RR[t]\\setminus \\RR$, constants $\\gamma, \\delta \\in \\RR^{\\neq 0}$, and $m, n\\in \\NN^{\\geq 1}$. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results in this chapter are sum-product type theorems for two polynomials, generalizing the classical result by Erd\\H os and Szemer\\'edi as well as a theorem by Shen. We also obtain similar results for $\\CC$, and from this deduce results for fields of characteristic $0$ and fields of large prime characteristic. We use tools from semialgebraic/o-minimal geometry to prove these results; exposition is provided on these methods to make them accessible to readers primarily concerned with combinatorics. In the second, together with Balogh and F\\\"uredi \\cite{BFR}, we combine two elementary proofs to show that the maximum size of a Sidon set of $\\{ 1, 2, \\ldots, n\\}$ is at most $\\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$ - the first non-constant improvement of the error term $n^{1/4}$ in this classical combinatorial number theory problem since 1969. This implies improvements in some related problems which are also discussed."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120122"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Souktik Roy"],"dc:subject":["Sum Product","O-minimality","Sidon Sets"],"dc:title":["Two topics in arithmetic combinatorics"],"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":"2026-07-22T22:24:56Z"}