{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120256"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120256","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Analytic and ergodic properties of certain classes of Euclidean algorithms","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":["Siskaki, Maria"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Boca, Florin","Zaharescu, Alexandru","Rosenblatt, Joseph","Berndt, Bruce","Athreya, Jayadev"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:57Z","subjects":["Euclidean Algorithms","Continued Fractions","Quadratic Irrationals","Kloosterman Sums","Gauss Conjecture","Perron-frobenius Operator","Farey Sequence","Pair Correlation"],"languages":["en","eng"],"rights":["Copyright 2023 Maria Siskaki"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120256","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Boca, Florin","Zaharescu, Alexandru","Rosenblatt, Joseph","Berndt, Bruce","Athreya, Jayadev"]},{"key":"dc:creator","label":"Author","values":["Siskaki, Maria"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2023-04-21"]},{"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":["Euclidean Algorithms","Continued Fractions","Quadratic Irrationals","Kloosterman Sums","Gauss Conjecture","Perron-frobenius Operator","Farey Sequence","Pair Correlation"]}]},{"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 Maria Siskaki"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120256"]}]},{"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, Maria Siskaki, accepted the attached license on 2023-04-12 at 11:16.","The student, Maria Siskaki, submitted this Dissertation for approval on 2023-04-12 at 11:26.","This Dissertation was approved for publication on 2023-04-21 at 08:10.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18963 on 2023-09-01 at 17:08:17","This thesis is concerned with results on analytic and ergodic properties of certain Continued Fraction Gauss maps arising from various Euclidean division algorithms. First, we study the distribution of the periodic points of the Even, Backwards, and Odd Gauss type shifts. These points coincide with certain classes of reduced quadratic irrationals, and, for the Regular Continued Fraction Gauss map, are known \\cite{Fa,Po} to be uniformly distributed with respect to the Gauss measure. Here, we adapt a number theoretical approach initiated in \\cite{KO} and fully developed in \\cite{Bo} to show that the $E$-, $B$-, and $O$-periodic points are uniformly distributed with the respect to the corresponding invariant measures. This approach ultimately relies on the application of Weil bounds on Kloosterman sums, and therefore provides an effective error in the final asymptotic formulas. It is worth noting that the invariant measures of the Even and Backwards Continued Fractions Gauss maps are infinite, making a Perron-Frobenius functional analytic approach as in \\cite{Fa,Po} more complex. Second, we study the Gauss-Kuzmin-L\\'evy problem for the Nearest Integer Continued Fraction Gauss map through a functional analytic approach. In particular, we provide effective formulas for the fact that the probability of landing in $[0,x]$ after successive applications of the NICF Gauss map $T$ is asymptotically equal to the $T$-invariant measure of $[0,x]$. The constant giving the speed of convergence in this effective formula is smaller than the (optimal) Wirsing constant for the Regular Continued Fraction Gauss map case. Finally, we study the local statistics for sequences of rationals involved in the Diophantine approximation of regular reduced quadratic irrationals. In particular, the pair correlation functions of Farey fractions with denominators $q$ satisfying $(q, m) = 1$, respectively $q \\equiv b \\mod{m}$ with $(b, m) = 1$, are shown to exist and are explicitly computed."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Analytic and ergodic properties of certain classes of Euclidean algorithms"]}]}],"canonical_facts":{"dc:contributor":["Boca, Florin","Zaharescu, Alexandru","Rosenblatt, Joseph","Berndt, Bruce","Athreya, Jayadev"],"dc:creator":["Siskaki, Maria"],"dc:date":["2023-05","2023-04-21"],"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, Maria Siskaki, accepted the attached license on 2023-04-12 at 11:16.","The student, Maria Siskaki, submitted this Dissertation for approval on 2023-04-12 at 11:26.","This Dissertation was approved for publication on 2023-04-21 at 08:10.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18963 on 2023-09-01 at 17:08:17","This thesis is concerned with results on analytic and ergodic properties of certain Continued Fraction Gauss maps arising from various Euclidean division algorithms. First, we study the distribution of the periodic points of the Even, Backwards, and Odd Gauss type shifts. These points coincide with certain classes of reduced quadratic irrationals, and, for the Regular Continued Fraction Gauss map, are known \\cite{Fa,Po} to be uniformly distributed with respect to the Gauss measure. Here, we adapt a number theoretical approach initiated in \\cite{KO} and fully developed in \\cite{Bo} to show that the $E$-, $B$-, and $O$-periodic points are uniformly distributed with the respect to the corresponding invariant measures. This approach ultimately relies on the application of Weil bounds on Kloosterman sums, and therefore provides an effective error in the final asymptotic formulas. It is worth noting that the invariant measures of the Even and Backwards Continued Fractions Gauss maps are infinite, making a Perron-Frobenius functional analytic approach as in \\cite{Fa,Po} more complex. Second, we study the Gauss-Kuzmin-L\\'evy problem for the Nearest Integer Continued Fraction Gauss map through a functional analytic approach. In particular, we provide effective formulas for the fact that the probability of landing in $[0,x]$ after successive applications of the NICF Gauss map $T$ is asymptotically equal to the $T$-invariant measure of $[0,x]$. The constant giving the speed of convergence in this effective formula is smaller than the (optimal) Wirsing constant for the Regular Continued Fraction Gauss map case. Finally, we study the local statistics for sequences of rationals involved in the Diophantine approximation of regular reduced quadratic irrationals. In particular, the pair correlation functions of Farey fractions with denominators $q$ satisfying $(q, m) = 1$, respectively $q \\equiv b \\mod{m}$ with $(b, m) = 1$, are shown to exist and are explicitly computed."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120256"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Maria Siskaki"],"dc:subject":["Euclidean Algorithms","Continued Fractions","Quadratic Irrationals","Kloosterman Sums","Gauss Conjecture","Perron-frobenius Operator","Farey Sequence","Pair Correlation"],"dc:title":["Analytic and ergodic properties of certain classes of Euclidean algorithms"],"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":"2026-07-22T22:24:57Z"}