Friendly Measures, Homogeneous Flows and Dirichlet’s Theorem
نویسندگان
چکیده
Let m,n be positive integers, and denote byMm,n the space of m×n matrices with real entries. Dirichlet’s Theorem (hereafter abbreviated by ‘DT’) on simultaneous diophantine approximation states that for any A ∈ Mm,n (viewed as a system of m linear forms in n variables) and for any T > 1 there exist q = (q1, . . . , qn) ∈ Z r {0} and p = (p1, . . . , pm) ∈ Z satisfying the following system of inequalities: ‖Aq− p‖ < 1/T and ‖q‖ ≤ T . (1.1)
منابع مشابه
Dirichlet’s Theorem on Diophantine Approximation and Homogeneous Flows
Given an m×n real matrix Y , an unbounded set T of parameters t = (t1, . . . , tm+n) ∈ R m+n + with ∑m i=1 ti = ∑n j=1 tm+j and 0 < ε ≤ 1, we say that Dirichlet’s Theorem can be ε-improved for Y along T if for every sufficiently large t ∈ T there are nonzero q ∈ Z and p ∈ Z such that { |Yiq− pi| < εe −ti , i = 1, . . . ,m |qj | < εe tm+j , j = 1, . . . , n (here Y1, . . . , Ym are rows of Y ). ...
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