Displaying similar documents to “The distribution of rational points close to a smooth manifold and Hausdorff dimension”

An extension of a theorem of Duffin and Schaeffer in Diophantine approximation

Faustin Adiceam (2014)

Acta Arithmetica

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Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended to the case where the numerators and the denominators of the rational approximants are related by a congruential constraint stronger than coprimality.

Rational points on the unit sphere

Eric Schmutz (2008)

Open Mathematics

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It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; r i = a i b i for some integers a i, b i.⊎ for all i , 0 ⩽ a i ⩽ b i ⩽ ( 32 1 / 2 l o g 2 n ε ) 2 l o g 2 n . One consequence...