Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

Sabrina Kombrink, Erin Pearse, Steffen Winter

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

A long-standing conjecture of Lapidus states that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. It was shown by Falconer and Lapidus (working independently but both using renewal theory) that nonlattice self-similar subsets of ℝ are Minkowski measurable, and the converse was shown by Lapidus and v. Frankenhuijsen a few years later, using complex dimensions. Around that time, Gatzouras used renewal theory to show that nonlattice self-similar subsets of ℝd that satisfy the open set condition are Minkowski measurable for d≥1. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous work have been removed.
Original languageEnglish
Pages (from-to)1049-1070
Number of pages22
JournalMathematische Zeitschrift
Volume283
Issue number3-4
Early online date27 Feb 2016
DOIs
Publication statusPublished - 1 Aug 2016

Keywords

  • Self-similar set
  • lattice and nonlattice
  • Minkowski dimension
  • Minkowski measurability
  • Minkowski content

Fingerprint

Dive into the research topics of 'Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable'. Together they form a unique fingerprint.

Cite this