Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results

TL;DR

This survey reviews recent advances in estimating the Hausdorff dimension of Furstenberg sets, extending classical results to generalized measures and fractal directions, including zero-dimensional cases.

Contribution

It introduces generalized Hausdorff measure estimates for Furstenberg sets, extending classical bounds and exploring fractal directions and zero-dimensional endpoints.

Findings

Generalized Hausdorff measure estimates for Furstenberg sets.

Extension of classical bounds to zero-dimensional cases.

Results on Furstenberg sets with fractal sets of directions.

Abstract

In this survey we collect and discuss some recent results on the so called "Furstenberg set problem", which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an interval poitning in that direction of some prescribed dimension. This problem is closely related to the "Kakeya needle problem". In this work we approach this problem from a more general point of view, in terms of generalized Hausdorff measures associated to dimension functions. We generalize the known results in terms of "logarithmic gaps" and obtain analogues to the classical estimates. Moreover, these analogues allow us to extend our results to the zero dimensional endpoint. We also obtain results about the dimension of a variation of Furstenberg sets defined for a fractal set of directions. We prove analogous inequalities reflecting the interplay between the size of the set of directions and the size of each fiber. This problem is also studied in the general scenario of Hausdorff measures.