ON THE HAUSDORFF MEASURE OF SHRINKING TARGET SETS ON SELF‐CONFORMAL SETS. Issue 4 (13th September 2021)

• Record Type: Journal Article
• Authors: Allen, Demi; Bárány, Balázs
• Is Part Of: Mathematika. Volume 67:Issue 4(2021)
• Page Start: 807
• Page End: 839
• Publication Date: 2021-09-13
• Subjects: 28A80 -- 37C45 (primary) -- 11J83 (secondary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12106
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: In this article, we study the Hausdorff measure of shrinking target sets on self‐conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995 [ Invent. Math . 119 (1) (1995), 175–198]. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work of Levesley, Salp, and Velani [ Math. Ann . 338 (1) (2007), 97–118], and recent work of Baker [ Math. Proc. Cambridge Philos. Soc . 167 (3) (2019), 567–597] .