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Marstrand's density theorem in the Heisenberg group
Author(s) -
Chousionis Vasilis,
Tyson Jeremy T.
Publication year - 2015
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdv056
Subject(s) - mathematics , heisenberg group , measure (data warehouse) , integer (computer science) , euclidean geometry , group (periodic table) , metric (unit) , radon measure , combinatorics , euclidean distance , set (abstract data type) , pure mathematics , discrete mathematics , locally compact space , geometry , quantum mechanics , operations management , physics , database , computer science , economics , programming language
We prove that if μ is a Radon measure on the Heisenberg group H n such that the densityΘ s ( μ , · ) , computed with respect to the Korányi metric d H , exists and is positive and finite on a set of positive μ measure, then s is an integer. The proof relies on an analysis of uniformly distributed measures on ( H n , d H ) . We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub‐Riemannian setting from its Euclidean counterpart.