Integral matrices of fixed rank
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- by Yonatan R. Katznelson PDF
- Proc. Amer. Math. Soc. 120 (1994), 667-675 Request permission
Abstract:
Asymptotic formulæare derived for the number of $n \times m$ matrices of fixed rank $k$ with rational integral coefficients that are contained in a Euclidean ball of radius $T$ in ${{\mathbf {R}}^{n \times m}}$. It is assumed that $n \geqslant m > k \geqslant 1$ are fixed, and the asymptotics are valid as $T$ tends to infinity. The methods used are elementary.References
- J. W. S. Cassels, An introduction to the geometry of numbers, Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin-New York, 1971. Second printing, corrected. MR 0306130
- W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), no. 1, 143–179. MR 1230289, DOI 10.1215/S0012-7094-93-07107-4
- Y. R. Katznelson, Singular matrices and a uniform bound for congruence groups of $\textrm {SL}_n(\textbf {Z})$, Duke Math. J. 69 (1993), no. 1, 121–136. MR 1201694, DOI 10.1215/S0012-7094-93-06906-2
- Wolfgang M. Schmidt, Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height, Duke Math. J. 35 (1968), 327–339. MR 224562
- Carl Ludwig Siegel, Lectures on the geometry of numbers, Springer-Verlag, Berlin, 1989. Notes by B. Friedman; Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter; With a preface by Chandrasekharan. MR 1020761, DOI 10.1007/978-3-662-08287-4
- Audrey Terras, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin, 1988. MR 955271, DOI 10.1007/978-1-4612-3820-1
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 667-675
- MSC: Primary 11H06; Secondary 11P21
- DOI: https://doi.org/10.1090/S0002-9939-1994-1169034-3
- MathSciNet review: 1169034