We study the Fibonacci Sets from the point of view of their quantity with respect to discrepancy and numerical integration. We give a Fourier analytic proof of the fact that symmetrized Fibonacci Set has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, allowing us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among the two dimensional point sets. Furthermore, with the help of Dedekind Sums, we find the L2 discrepancy of rational approximation for the general irrational lattice and characterize the rational lattices for which the L2 discrepancy are optimal. We also introduce quartered Lp discrepancy and prove non-symmetrized Fibonacci Sets has optimal quartered Lp discrepancy.
","abstract_html":"<p> We study the Fibonacci Sets from the point of view of their quantity with respect to discrepancy and numerical integration. We give a Fourier analytic proof of the fact that symmetrized Fibonacci Set has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, allowing us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among the two dimensional point sets. Furthermore, with the help of Dedekind Sums, we find the L2 discrepancy of rational approximation for the general irrational lattice and characterize the rational lattices for which the L2 discrepancy are optimal. We also introduce quartered Lp discrepancy and prove non-symmetrized Fibonacci Sets has optimal quartered Lp discrepancy.</p>","abstract_has_math":false,"creators":["Yu, Rui"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Vladimir Temlyakov"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:37Z","subjects":["Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["© 2012, Rui Yu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1624","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Vladimir Temlyakov"]},{"key":"dc:creator","label":"Author","values":["Yu, Rui"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Rui Yu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1624"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We study the Fibonacci Sets from the point of view of their quantity with respect to discrepancy and numerical integration. We give a Fourier analytic proof of the fact that symmetrized Fibonacci Set has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, allowing us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among the two dimensional point sets. Furthermore, with the help of Dedekind Sums, we find the L2 discrepancy of rational approximation for the general irrational lattice and characterize the rational lattices for which the L2 discrepancy are optimal. We also introduce quartered Lp discrepancy and prove non-symmetrized Fibonacci Sets has optimal quartered Lp discrepancy.
"]},{"key":"dc:title","label":"Title","values":["Fibonacci Sets In Discrepancy Theory and Numerical Integration"]}]}],"canonical_facts":{"dc:contributor":["Vladimir Temlyakov"],"dc:creator":["Yu, Rui"],"dc:description.abstract":["We study the Fibonacci Sets from the point of view of their quantity with respect to discrepancy and numerical integration. We give a Fourier analytic proof of the fact that symmetrized Fibonacci Set has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, allowing us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among the two dimensional point sets. Furthermore, with the help of Dedekind Sums, we find the L2 discrepancy of rational approximation for the general irrational lattice and characterize the rational lattices for which the L2 discrepancy are optimal. We also introduce quartered Lp discrepancy and prove non-symmetrized Fibonacci Sets has optimal quartered Lp discrepancy.
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1624"],"dc:rights":["© 2012, Rui Yu"],"dc:subject":["Mathematics","Physical Sciences and Mathematics"],"dc:title":["Fibonacci Sets In Discrepancy Theory and Numerical Integration"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:38:37Z"}