#
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.