Math. Model. Nat. Phenom.
Volume 9, Number 5, 2014Spectral problems
|Page(s)||148 - 169|
|Published online||17 July 2014|
Theory of Dimension for Large Discrete Sets and Applications
1 Department of Mathematics, University of Rochester,
Rochester, NY 14627
2 Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
3 Department of Mathematics, Michigan State University, East Lansing MI 48824
Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 11H99 / 28A75 / 31A15 / 31B15 / 52C10
Key words: Falconer and Erdös distance problems / Hausdorff and Minkowski dimensions / geometric measure theory / adaptability
© EDP Sciences, 2014
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