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On the Convergence of Mean Values Over Lattices

Published online by Cambridge University Press:  20 November 2018

Wolfgang Schmidt
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Recently C. A.Rogers (2, Theorem 4) proved the following theorem which applies to many problems in geometry of numbers:

Let f(X1,X2, … , Xk) be a non-negative B or el-measurable function in the nk-dimensional space of points (X1,X2. Further, let Λo be the fundamental lattice, Ω a linear transformation of determinant 1, F a fundamental region in the space of linear transformations of determinant 1, defined with respect to the subgroup of unimodular transformations and μ(Ω) the invariant measure1 on the space of linear transformations of determinant 1 in Rn.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 103 - 110
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Butson, A. T. and Stewart, B. M., Systems of linear congruences, Can. J. Math., 7 (1955), 358-368.Google Scholar
2. Rogers, C. A., Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.Google Scholar
3. Rogers, C. A., A single integral inequality, to appear in the Journal of the London Math. Soc.Google Scholar
4. Schmidt, W., Mittelwerte ilber Gitter, Monatshefte f. Math., 61 (1957), 000-000.Google Scholar
5. Siegel, C. L., A mean value theorem in geometry of numbers, Annals of Math., 46 (1945), 340-347.Google Scholar