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AN EXTENSION OF A THEOREM OF HLAWKA

Published online by Cambridge University Press:  29 April 2010

Martin Moskowitz  and
Richard Sacksteder
Martin Moskowitz
Affiliation:
Mathematics Ph.D. Program, The Graduate Center, City University of New York, New York, NY 10016, U.S.A. (email: martin.moskowitz@gmail.com)
Richard Sacksteder
Affiliation:
Mathematics Ph.D. Program, The Graduate Center, City University of New York, New York, NY 10016, U.S.A.
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Abstract

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 Vn=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 Vn can be expressed in terms of the Riemann zeta function by As a consequence, let D be a domain of a sufficiently regular set in ℝ2n. Then:

  1. (i) if vol(D)>Vn, then some lattice in ℝ2n contains a non-zero point of D;

  2. (ii) if vol(D)<Vn, then some lattice in ℝ2n contains only the zero point of D;

  3. (iii) if D is star-shaped about the origin and vol(D)<ζ(2n)Vn, then some lattice in ℝ2n 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.

Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 203 - 216
Copyright
Copyright © University College London 2010

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