Theory of Dimension for Large Discrete Sets and Applications

Alex Iosevich, M. Rudnev, Ignacio Uriarte-Tuero

Type: Article

Publication Date: 2014-01-01

Citations: 20

DOI: https://doi.org/10.1051/mmnp/20149510

Abstract

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdős and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.

Locations

Mathematical Modelling of Natural Phenomena - View - PDF
arXiv (Cornell University) - View - PDF
Springer Link (Chiba Institute of Technology) - View - PDF

Similar Works

Action	Title	Year	Authors
+	Theory of dimension for large discrete sets and applications	2007	Alex Iosevich Misha Rudnev Ignacio Uriarte-Tuero
+ PDF Chat	The Hausdorff dimension of sets arising in metric Diophantine approximation	1994	H. Dickinson
+	Hausdorff dimension and distance sets	1994	Jean Bourgain
+	Introduction to Hausdorff measure and dimension	2024
+ PDF Chat	Some Problems and Ideas of Erdős in Analysis and Geometry	2013	R. Daniel Mauldin
+ PDF Chat	Fractal Dimension and Lower Bounds for Geometric Problems	2021	Anastasios Sidiropoulos Kritika Singhal Vijay Sridhar
+	Lower bounds for Hausdorff dimension	1999	В. И. Берник M. M. Dodson
+	Hausdorff dimension: Techniques and applications	2001	Peter Mörters Yuval Peres
+	Hausdorff Measure and Isodiametric Inequalities	2005	Zhou Lingnan
+	Algorithmic interpretations of fractal dimension	2017	Anastasios Sidiropoulos Vijay Sridhar
+	On the Erdős–Falconer distance problem for two sets of different size in vector spaces over finite fields	2013	Rainer Dietmann
+	Finite point configurations and projection theorems in vector spaces over finite fields	2010	Jeremy Chapman
+	Hausdorff Measure and Dimension	2022
+	Bounding the Dimension of Points on a Line	2016	Neil Lutz D. M. Stull
+	Bounding the dimension of points on a line	2020	Neil Lutz D. M. Stull
+	Hausdorff dimension bounds	2014	Richard Evan Schwartz
+ PDF Chat	Hausdorff dimension and p-adic diophantine approximation	1999	H. Dickinson M. M. Dodson Jun Yuan
+	K-distance sets, Falconer conjecture, and discrete analogs	2003	Alex Iosevich Izabella Łaba
+	Algorithmic Interpretations of Fractal Dimension	2017	Anastasios Sidiropoulos Vijay Sridhar
+	The Dimensions of Hyperspaces	2020	Jack H. Lutz Neil Lutz Elvira Mayordomo

Works That Cite This (20)

Action	Title	Year	Authors
+	Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤ<sup><i>n</i></sup>	2015	Driss Essouabri Ben Lichtin
+	Unit distance problems	2015	Daniel M. Oberlin Richard Oberlin
+	Sharpness of Falconer's estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains	2010	Alex Iosevich Steven Senger
+	EXPLORATIONS OF THE ERD } OS-FALCONER DISTANCE PROBLEM AND RELATED APPLICATIONS	2011	Steven Senger Alex Iosevich
+	Upper bounds on pairs of dot products	2015	Daniel W. Barker Steven Senger
+ PDF Chat	Additive and geometric transversality of fractal sets in the integers	2024	Daniel Glasscock Joel Moreira Florian K. Richter
+ PDF Chat	On Falconer’s distance set problem in the plane	2019	Larry Guth Alex Iosevich Yumeng Ou Hong Wang
+	Thick self similar sets are asymptotically generic	2018	Driss Essouabri Ben Lichtin
+ PDF Chat	Visibility phenomena in hypercubes	2023	Jayadev S. Athreya Cristian Cobeli Alexandru Zaharescu
+	Dot product chains	2020	Shelby Kilmer Caleb Marshall Steven Senger

Works Cited by This (15)

Action	Title	Year	Authors
+	Potential Theory in the Complex Plane	1995	Thomas Ransford
+	None	1999	Thomas Wolff
+	Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability	1995	Pertti Mattila
+	On the Erdos distinct distance problem in the plane	2010	Larry Guth Nets Hawk Katz
+	Hausdorff dimension and distance sets	1994	Jean Bourgain
+	Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets	1987	Pertti Mattila
+	Near optimal bounds for the Erdős distinct distances problem in high dimensions	2008	József Solymosi Van H. Vu
+ PDF Chat	K-distance Sets, Falconer Conjecture, and Discrete Analogs	2005	Alex Iosevich Izabella Łaba
+ PDF Chat	On the Hausdorff dimensions of distance sets	1985	K. J. Falconer
+ PDF Chat	Bounds on arithmetic projections, and applications to the Kakeya conjecture	1999	Nets Hawk Katz Terence Tao