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Theory of Dimension for Large Discrete Sets and Applications

Published online by Cambridge University Press:  17 July 2014

A. Iosevich ,
M. Rudnev  and
I. Uriarte-Tuero
A. Iosevich
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627
M. Rudnev
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
I. Uriarte-Tuero*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing MI 48824
*
⋆ Corresponding author. E-mail: ignacio@math.msu.edu
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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.

Keywords

Falconer and Erdös distance problemsHausdorff and Minkowski dimensionsgeometric measure theoryadaptability
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 148 - 169
Copyright
© EDP Sciences, 2014

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