Theory of dimension for large discrete sets and applications

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 Erdos and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.

27 pages

Cited by in corpus (5)

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• On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators
• On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry
• On sets of directions determined by subsets of ${\mathbb{R}}^d$
• The Erdös-Falconer distance problem on the unit sphere in vector spaces over finite fields