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AN EXTENSION OF A THEOREM OF HLAWKA
Author(s) -
Moskowitz Martin,
Sacksteder Richard
Publication year - 2010
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579310000392
Subject(s) - mathematics , combinatorics , lattice (music) , zero (linguistics) , physics , philosophy , linguistics , acoustics
This paper extends Hlawka's theorem (from the point of view of Siegel and Weil) on SL ( n ,ℝ)/ SL ( n ,ℤ) to Sp ( n ,ℝ)/ Sp ( n ,ℤ). Namely, if V n = vol ( Sp ( n ,ℝ)/ Sp ( n ,ℤ), where the measure is the Sp ( n ,ℝ)‐invariant measure on Sp ( n ,ℝ)/ Sp ( n ,ℤ), then V n can be expressed in terms of the Riemann zeta function byV n = 1 2∏ k = 1 n ζ ( 2 k ) .As a consequence, let D be a domain of a sufficiently regular set in ℝ 2 n . Then: (i) if vol ( D )> V n , then some lattice in ℝ 2 n contains a non‐zero point of D ; (ii) if vol ( D )< V n , then some lattice in ℝ 2 n contains only the zero point of D ; (iii) if D is star‐shaped about the origin and vol ( D )< ζ (2 n ) V n , then some lattice in ℝ 2 n contains only the zero point of D . At the same time, we also obtain unity with the “classical” SL ( n ,ℝ)/ SL ( n ,ℤ) case.