Rectangular Shrinking Targets on Self-Similar Carpets
Dynamical Systems
2025-10-10 v2
Abstract
Since the introduction of the shrinking target problem by Hill and Velani in 1995 there has been a surge of interest in the area. In this paper we consider the case where the target is a rectangle, rather than a ball, and the underlying space is a self-similar carpet. We calculate the exact Hausdorff dimension of the resulting shrinking target set. Interestingly the Hausdorff dimension depends on the centre of the target, a condition uncommon in most other shrinking target type problems. This extends a theorem of Wang and Wu [Theorem 12.1, Math. Ann. 2021].
Cite
@article{arxiv.2510.04875,
title = {Rectangular Shrinking Targets on Self-Similar Carpets},
author = {Demi Allen and Thomas Jordan and Benjamin Ward},
journal= {arXiv preprint arXiv:2510.04875},
year = {2025}
}
Comments
31 pages. A typo corrected in the statement of Theorem 1.1