A note on the Hausdorff dimension of uniformly non flat sets with plenty of big projections

TL;DR

This paper demonstrates that sets with many large projections and non-flat geometry have high Hausdorff dimension, using recent theorems and establishing connections to harmonic capacities.

Contribution

It introduces a new Analyst's Travelling Salesman Theorem for sets with big projections and links uniform non-flatness to large Hausdorff dimension.

Findings

Sets with plenty of big projections have an Analyst's TSP.

Uniformly non-flat sets with big projections have large Hausdorff dimension.

Corollary on analytic and Lipschitz harmonic capacities.

Abstract

Using a recent result of Orponen (Invent. math. '21), we show that sets with plenty of big projections (PBP) admit an Analyst's Travelling Salesman Theorem. We then show that sets with PBP which are uniformly non-flat (or wiggly) have large Hausdorff dimension. We also obtain a corollary on analytic/Lipschitz harmonic capacities.