Lattice orbit distribution on ℝ2
2009 ◽
Vol 30
(4)
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pp. 1201-1214
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Keyword(s):
AbstractWe study the distribution on ℝ2 of the orbit of a vector under the linear action of SL(2,ℤ). Let Ω⊂ℝ2 be a compact set and x∈ℝ2. Let N(k,x) be the number of matrices γ∈SL(2,ℤ) such that γ(x)∈Ω and ‖γ‖≤k, k=1,2,…. If Ω is a square, we prove the existence of an absolute error term for N(k,x), as k→∞, for almost every x, which depends on the Diophantine property of the ratio of the coordinates of x. Our approach translates the question into a Diophantine approximation counting problem which provides the absolute error term. The asymptotical behaviour of N(k,x) is also obtained using ergodic theory.
2013 ◽
Vol 378
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pp. 459-465
Keyword(s):
2005 ◽
Vol 23
(4)
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pp. 1093-1101
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Keyword(s):
2021 ◽
Vol 62
(9)
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pp. 1181-1188
Keyword(s):
Keyword(s):
2016 ◽
Vol 11
(5)
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pp. 1003-1016
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2014 ◽
Vol 590
◽
pp. 651-655
2012 ◽
Vol 239-240
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pp. 344-347
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