Improving dimension estimates for Furstenberg-type sets

TL;DR

This paper establishes new lower bounds on the Hausdorff dimension of Furstenberg-type sets and their generalizations, advancing understanding of their geometric measure properties.

Contribution

It introduces novel lower bounds for the Hausdorff dimension of Furstenberg sets and extends these results to generalized sets with doubling dimension functions.

Findings

Lower bounds on Hausdorff dimension of Furstenberg sets

Extension to generalized Furstenberg sets with doubling dimension functions

Lower bounds for zero-dimensional Furstenberg sets under growth conditions

Abstract

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.