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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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Attribution-NonCommercial-NoDerivs 2.5 Canada