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Finite configurations in sparse sets Chan, Vincent
Abstract
We prove a result which adds to the study of continuous analogues of Szemerédi-type problems. Let E ⊆ ℝⁿ be a Lebesgue-null set of Hausdorff dimension α, k, m be integers satisfying a suitable relationship, and {B₁,…, Bk} be n × (m − n) matrices. We prove that if the set of matrices Bi are non-degenerate in a particular sense, α is sufficiently close to n, and if E supports a probability measure satisfying certain dimensionality and Fourier decay conditions, then E contains a k-point configuration of the form {x + B₁y,…,x + Bky}. In particular, geometric configurations such as collinear triples, triangles, and parallelograms are contained in sets satisfying the above conditions.
Item Metadata
Title |
Finite configurations in sparse sets
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2014
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Description |
We prove a result which adds to the study of continuous analogues of Szemerédi-type problems. Let E ⊆ ℝⁿ be a Lebesgue-null set of Hausdorff dimension α, k, m be integers satisfying a suitable relationship, and {B₁,…, Bk} be n × (m − n) matrices. We prove that if the set of matrices Bi are non-degenerate in a particular sense, α is sufficiently close to n, and if E supports a probability measure satisfying certain dimensionality and Fourier decay conditions, then E contains a k-point configuration of the form {x + B₁y,…,x + Bky}. In particular, geometric configurations such as collinear triples, triangles, and parallelograms are contained in sets satisfying the above conditions.
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Genre | |
Type | |
Language |
eng
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Date Available |
2014-04-10
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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DOI |
10.14288/1.0166905
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2014-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivs 2.5 Canada