A DICHOTOMY LAW FOR THE DIOPHANTINE PROPERTIES IN ‐DYNAMICAL SYSTEMS

2016/01/01 by Michael Coons, Mumtaz Hussain, Bao-Wei Wang · 8 citations
Mathematics · #Diophantine approximation #Diophantine equation #Divergence (linguistics) #Function (biology) #Hausdorff measure #Hausdorff space #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Measure (data warehouse) #Zero (linguistics) #advanced mathematical theories #math.DS #math.NT #msc:11J83 #msc:11K16 #msc:11K60

paper · pdf · doi:10.1112/s0025579316000085

published in Mathematika 62(3), 884-897 (Wiley) · Accepted for publication in Mathematika

openalex publication_date 2016/01/01 · arxiv created 2016/04/04 · arxiv updated 2016/05/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Let be a real number and define the -transformation on by . Further, define and where is a positive function such that as . In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

Citations

• On theβ-expansions of real numbers
• Representations for real numbers and their ergodic properties
• A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures