{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98203"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98203","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Correlations of sequences modulo one and statistics of geometrical objects associated to visible points","abstract":"This thesis is divided into two major topics. In the first, we study the topic of distribution of sequences modulo one. In particular, we look at the spacing distributions between members of rational valued sequences modulo one. We come up with examples of many such sequences which behave as randomly chosen numbers from the unit interval. These include examples from the class of exponentially as well as sub-exponentially growing sequences. In the second part, we examine distribution questions for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical interpretation in the form of Ford circles. We study the distribution of moments of various geometrical parameters associated to these objects by giving asymptotic formulas employing tools from analytic number theory.","abstract_html":"This thesis is divided into two major topics. In the first, we study the topic of distribution of sequences modulo one. In particular, we look at the spacing distributions between members of rational valued sequences modulo one. We come up with examples of many such sequences which behave as randomly chosen numbers from the unit interval. These include examples from the class of exponentially as well as sub-exponentially growing sequences. In the second part, we examine distribution questions for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical interpretation in the form of Ford circles. We study the distribution of moments of various geometrical parameters associated to these objects by giving asymptotic formulas employing tools from analytic number theory.","abstract_has_math":false,"creators":["Chaubey, Sneha"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zaharescu, Alexandru","Boca, Florin","Hildebrand, A. J.","Robles, Nicolas"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T17:45:18Z","date_published":"2017-09-29T17:45:18Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Pair correlation","Riemann zeta","Visible lattice points"],"languages":["en"],"rights":["Copyright 2017 Sneha Chaubey"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98203","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zaharescu, Alexandru","Boca, Florin","Hildebrand, A. 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In the first, we study the topic of distribution of sequences modulo one. In particular, we look at the spacing distributions between members of rational valued sequences modulo one. We come up with examples of many such sequences which behave as randomly chosen numbers from the unit interval. These include examples from the class of exponentially as well as sub-exponentially growing sequences. In the second part, we examine distribution questions for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical interpretation in the form of Ford circles. We study the distribution of moments of various geometrical parameters associated to these objects by giving asymptotic formulas employing tools from analytic number theory.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2019-08-01","The student, Sneha Chaubey, accepted the attached license on 2017-07-10 at 00:37.","The student, Sneha Chaubey, submitted this Dissertation for approval on 2017-07-10 at 00:48.","This Dissertation was approved for publication on 2017-07-10 at 14:55.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11355 on 2017-09-29 at 10:46:47","Made available in DSpace on 2017-09-29T17:45:18Z (GMT). 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J.","Robles, Nicolas"],"dc:creator":["Chaubey, Sneha"],"dc:date":["2017-09-29T17:45:18Z","2020-03-03T10:15:22Z","2017-07-10","2017-08"],"dc:description":["This thesis is divided into two major topics. In the first, we study the topic of distribution of sequences modulo one. In particular, we look at the spacing distributions between members of rational valued sequences modulo one. We come up with examples of many such sequences which behave as randomly chosen numbers from the unit interval. These include examples from the class of exponentially as well as sub-exponentially growing sequences. In the second part, we examine distribution questions for certain geometrical objects, for example, Farey-Ford and generalized Farey-Ford polygons and Farey-Ford parabolas associated to visible lattice points. As the names suggest, these objects are constructed based on the relation between visible points/Farey fractions and their geometrical interpretation in the form of Ford circles. We study the distribution of moments of various geometrical parameters associated to these objects by giving asymptotic formulas employing tools from analytic number theory.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2019-08-01","The student, Sneha Chaubey, accepted the attached license on 2017-07-10 at 00:37.","The student, Sneha Chaubey, submitted this Dissertation for approval on 2017-07-10 at 00:48.","This Dissertation was approved for publication on 2017-07-10 at 14:55.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11355 on 2017-09-29 at 10:46:47","Made available in DSpace on 2017-09-29T17:45:18Z (GMT). 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