TY - JOUR

T1 - Dimensional lower bounds for Falconer type incidence theorems

AU - DeWitt, Jonathan

AU - Ford, Kevin

AU - Goldstein, Eli

AU - Miller, Steven J.

AU - Moreland, Gwyneth

AU - Palsson, Eyvindur A.

AU - Senger, Steven

N1 - Funding Information:
Let 1 ≤ k ≤ d and consider a subset E ⊂ ℝ d . In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k -simplices in E (that is, noncongruent point configurations of k + 1 points from E ) to have positive Lebesgue measure. This generalizes the k = 1 case, the well-known Falconer distance problem and a major open problem in geometric measure theory. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We establish a dimensional lower threshold of d + 1 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d+1}{2}$$\end{document} on incidence theorems for k -simplices where k ≤ d ≤ 2 k + 1 by generalizing an example of Mattila. We also prove a dimensional lower threshold of d + 1 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d+1}{2}$$\end{document} on incidence theorems for triangles in a convex setting in every dimension greater than 3. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation. publisher-imprint-name Hebrew University Magnes Press article-contains-esm No article-numbering-style Unnumbered article-registration-date-year 2019 article-registration-date-month 10 article-registration-date-day 8 article-toc-levels 0 journal-product ArchiveJournal numbering-style Unnumbered article-grants-type Regular metadata-grant OpenAccess abstract-grant OpenAccess bodypdf-grant Restricted bodyhtml-grant Restricted bibliography-grant Restricted esm-grant OpenAccess online-first true pdf-file-reference BodyRef/PDF/11854_2019_Article_56.pdf target-type OnlinePDF article-type OriginalPaper journal-subject-primary Mathematics journal-subject-secondary Analysis journal-subject-secondary Functional Analysis journal-subject-secondary Dynamical Systems and Ergodic Theory journal-subject-secondary Abstract Harmonic Analysis journal-subject-secondary Partial Differential Equations journal-subject-collection Mathematics and Statistics open-access false The first, third, fourth and fifth authors were supported in part by National Science Foundation grants DMS1265673, DMS1561945 and DMS1347804. The second-listed author was supported in part by National Science Foundation grant DMS1501982 and the sixth-listed author was supported supported in part by Simons Foundation Grant #360560. The authors thank an anonymous referee for suggestions that significantly improved the paper.
Publisher Copyright:
© 2019, The Hebrew University of Jerusalem.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - Let 1 ≤ k ≤ d and consider a subset E ⊂ ℝd. In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k + 1 points from E) to have positive Lebesgue measure. This generalizes the k = 1 case, the well-known Falconer distance problem and a major open problem in geometric measure theory. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We establish a dimensional lower threshold of d+12 on incidence theorems for k-simplices where k ≤ d ≤ 2k + 1 by generalizing an example of Mattila. We also prove a dimensional lower threshold of d+12 on incidence theorems for triangles in a convex setting in every dimension greater than 3. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

AB - Let 1 ≤ k ≤ d and consider a subset E ⊂ ℝd. In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k + 1 points from E) to have positive Lebesgue measure. This generalizes the k = 1 case, the well-known Falconer distance problem and a major open problem in geometric measure theory. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We establish a dimensional lower threshold of d+12 on incidence theorems for k-simplices where k ≤ d ≤ 2k + 1 by generalizing an example of Mattila. We also prove a dimensional lower threshold of d+12 on incidence theorems for triangles in a convex setting in every dimension greater than 3. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

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U2 - 10.1007/s11854-019-0056-0

DO - 10.1007/s11854-019-0056-0

M3 - Article

AN - SCOPUS:85074535425

VL - 139

SP - 143

EP - 154

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -