Effective methods in measure and dimension

August 15 to August 19, 2022

at the

American Institute of Mathematics, San Jose, California

organized by

Denis R. Hirschfeldt, Joseph S. Miller, and Theodore A. Slaman

Original Announcement

This workshop will be devoted to effective approaches to geometric measure theory. Algorithmic Randomness and Effective Descriptive Set Theory provide a novel perspective to many concepts that are classically measure-based, such as randomness and Hausdorff dimension. A core feature of this perspective is that it links the geometric properties of a set to the logical and computational complexity of its points. The new techniques have been successfully applied to extend previous results in the area to larger classes of sets (beyond analytic), and also to shed new light on why in other cases such an extension is impossible. Examples include Marstrand's projection theorem and the capacitability of sets of real numbers.

The goal of this workshop is to bring together researchers with expertise in computability, set theory, geometric measure theory, and related areas to further develop these new approaches. The following topics seem particularly promising:

Strong measure vs strong dimension.
Besicovitch showed that the continuum hypothesis implies that there exist sets with a ``strong'' dimension property: that is they have non-$\sigma$-finite linear measure and have $\mathcal{H}^\psi$-measure zero for any gauge function $\psi$ higher than linear. Such sets are necessarily non-analytic but can be co-analytic if $V=L$. More generally, the failure of the Borel Conjecture for sets of strong measure zero implies that there are non-$\sigma$-finite sets of strong linear dimension. Does the assumption of $AD^{L[\mathbb{R}]}$ imply that there are no such sets in $L[\mathbb{R}]$? Is their existence equivalent to the failure of the Borel conjecture?
Point-to-set principles.
There is a tight correspondence between the classical Hausdorff dimension of a set in Euclidean space and the effective Hausdorff dimension of its points. Questions about the Hausdorff dimension of a set can then be translated into questions about the relative randomness and effective dimension of single points. To what extent does this principle extend to general gauge functions and to metric spaces other than Euclidean?
Towards a local notion of Fourier dimension.
The point-to-set principle for Hausdorff dimension can be seen as an instance of a local-global correspondence for Hausdorff dimension of measures. For the Fourier dimension of a measure (defined based on the asymptotic behavior of the Fourier-Stieltjes coefficients of a measure - a global property) such a correspondence is currently not known.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Inequalities for entropies and dimensions

by Alexander Shen