Counting Singular Matrices with Primitive Row Vectors |
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Authors: | Igor Wigman |
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Institution: | (1) Tel Aviv University, Israel |
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Abstract: | We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n × n 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 T![thinsp](/content/bla52jxw72855la6/xxlarge8201.gif) ![rarr](/content/bla52jxw72855la6/xxlarge8594.gif) ![thinsp](/content/bla52jxw72855la6/xxlarge8201.gif) , the number is asymptotic to
for n![thinsp](/content/bla52jxw72855la6/xxlarge8201.gif) ![ge](/content/bla52jxw72855la6/xxlarge8805.gif) 3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt. |
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Keywords: | 2000 Mathematics Subject Classification: 11H06 |
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