Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

TY - JOUR

T1 - Diophantine approximation in Kleinian groups

T2 - singular, extremal, and bad limit points

AU - Beresnevich, Victor

AU - Ghosh, Anish

AU - Simmons, David Samuel

AU - Velani, Sanju

N1 - ©2018 The Author(s)

PY - 2018/10/1

Y1 - 2018/10/1

N2 - The overall aim of this note is to initiate a ‘manifold’ theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural ‘manifold’ strengthening of Sullivan's logarithmic law for geodesics.

AB - The overall aim of this note is to initiate a ‘manifold’ theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural ‘manifold’ strengthening of Sullivan's logarithmic law for geodesics.

KW - 11J83

KW - 11K60 (primary)

KW - 30F40 (secondary)

UR - http://www.scopus.com/inward/record.url?scp=85054294228&partnerID=8YFLogxK

U2 - 10.1112/jlms.12133

DO - 10.1112/jlms.12133

M3 - Article

SN - 1469-7750

VL - 98

SP - 306

EP - 328

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 2

ER -