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Counting Singular Matrices with Primitive Row Vectors
Authors:Igor Wigman
Institution:(1) Tel Aviv University, Israel
Abstract:We solve an asymptotic problem in the geometry of numbers, where we count the number of singular nthinsp×thinspn matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as Tthinsprarrthinspinfin, the number is asymptotic to ${(n-1)u_n \over \zeta (n) \zeta(n-1)^n}T^{n^2-n}\log (T)$ for nthinspgethinsp3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt.
Keywords:2000 Mathematics Subject Classification: 11H06
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