Computability and Set Theoretic Aspects of Hausdorff Dimension

About this item

Description:	Theodore Slaman (University of California, Berkeley) 08/06/2022 Programme: SASW09 SemId: 36176


Created:	2022-06-09 19:59
Collection:	International conference on computability, complexity and randomness
Publisher:	Theodore Slaman
Copyright:	Isaac Newton Institute
Language:	eng (English)


Abstract:	The Hausdorff Dimension of a set of real numbers A is a numerical indication of the geometric fullness of A. Sets of positive measure have dimension 1, but there are null sets of every possible dimension between 0 and 1. Effective Hausdorff Dimension is a variant which incorporates computability-theoretic considerations. By work of Jack and Neil Lutz, Elvira Mayordomo, and others, there is a direct connection between the the effective Hausdorff dimensions of the elements of a set A and the Hausdorff dimension of A itself. We will describe how this point-to-set principle works, how allows for novel approaches to classical problems and further lines of investigation in Geometric Measure Theory.

Abstract:

The Hausdorff Dimension of a set of real numbers A is a numerical indication of the geometric
fullness of A. Sets of positive measure have dimension 1, but there are null sets of every possible
dimension between 0 and 1.
Effective Hausdorff Dimension is a variant which incorporates computability-theoretic
considerations. By work of Jack and Neil Lutz, Elvira Mayordomo, and others, there is a direct
connection between the the effective Hausdorff dimensions of the elements of a set A and the
Hausdorff dimension of A itself. We will describe how this point-to-set principle works, how allows
for novel approaches to classical problems and further lines of investigation in Geometric Measure
Theory.