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- W2913060592 abstract "In this paper, all numbers are real and all radicals are positive. Let f ( x, y ) = ax 2 + bxy + cy 2 be an indefinite quadratic form, and let d = Δ 2 = b 2 − 4 ac , where Δ > 0. A well-known theorem of Minkowski states that, if ( x 0 , y 0 ) is any pair of numbers, then there exists a pair ( x, y ), x ≡ x 0 (mod 1), y ≡ y 0 (mod 1), say ( x, y ) ≡ ( x 0 , y 0 ) (mod 1), such that for k ≥ ¼ , and Davenport has shown how this result may be improved if we know a value assumed by f ( x, y ) for coprime integral values, ( x, y ) = ( m, n ) ≠ (0, 0). In this paper, we discuss the more general inequality where R and S are constants, and use the method developed in the first paper of this series to obtain a sharper and more general theorem than Davenport's. We give an application to the theory of real Euclidean quadratic fields and to a problem in Diophantine approximation discussed by Khintchine." @default.
- W2913060592 created "2019-02-21" @default.
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- W2913060592 date "1948-04-01" @default.
- W2913060592 modified "2023-10-17" @default.
- W2913060592 title "The lattice properties of asymmetric hyperbolic regions" @default.
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- W2913060592 doi "https://doi.org/10.1017/s0305004100024117" @default.
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