{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/151380"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/151380","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Application of high-low method to distance problems","abstract":"In this thesis, I used the high-low method to study incidence problems arising form variants of the distance set conjectures. In the complex space, my collaborator Sarah Tammen and I proved analogues of the incidence estimates of Guth, Solomon and Wang \\cite{WellSpacedTubes} for tubes obeying strong spacing conditions, and we used one of our new estimates to resolve a discretized variant of Falconer's distance set problem in C². In the real space, I proved an incidence estimate for special boxes in R⁶, and used the estimate to derive a bound for a discretized variant of Falconer's problem in R³.","abstract_html":"In this thesis, I used the high-low method to study incidence problems arising form variants of the distance set conjectures. In the complex space, my collaborator Sarah Tammen and I proved analogues of the incidence estimates of Guth, Solomon and Wang \\cite{WellSpacedTubes} for tubes obeying strong spacing conditions, and we used one of our new estimates to resolve a discretized variant of Falconer's distance set problem in C². In the real space, I proved an incidence estimate for special boxes in R⁶, and used the estimate to derive a bound for a discretized variant of Falconer's problem in R³.","abstract_has_math":false,"creators":["Zhang, Lingxian"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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In the complex space, my collaborator Sarah Tammen and I proved analogues of the incidence estimates of Guth, Solomon and Wang \\cite{WellSpacedTubes} for tubes obeying strong spacing conditions, and we used one of our new estimates to resolve a discretized variant of Falconer's distance set problem in C². In the real space, I proved an incidence estimate for special boxes in R⁶, and used the estimate to derive a bound for a discretized variant of Falconer's problem in R³."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Application of high-low method to distance problems"]}]}],"canonical_facts":{"dc:contributor.advisor":["Guth, Larry"],"dc:contributor.department":["Massachusetts Institute of Technology. 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In the real space, I proved an incidence estimate for special boxes in R⁶, and used the estimate to derive a bound for a discretized variant of Falconer's problem in R³."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/151380"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"dc:rights.uri":["https://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Application of high-low method to distance problems"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:22:17Z"}