The distribution of rational points close to a smooth manifold and Hausdorff dimension

Vasilij I. Bernik; Ella I. Kovalevskaya

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

Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation.

M.M. Dodson (1992)

Journal für die reine und angewandte Mathematik

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Twisted inhomogeneous Diophantine approximation and badly approximable sets

Stephen Harrap (2012)

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

Faustin Adiceam (2014)

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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.

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Bryna Kra, Jorg Schmeling (2002)

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Yann Bugeaud, Nicolas Chevallier (2006)

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Makoto Nagata (2003)

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Richard Hill, Sanju L. Velani (1997)

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A metric theorem for restricted Diophantine approximation in positive characteristic

Simon Kristensen (2006)

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Sets of exact approximation order by rational numbers III

Yann Bugeaud, Carlos Gustavo Moreira (2011)

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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; $r_{i} = \frac{a_{i}}{b_{i}}$ for some integers a i, b i.⊎ for all $i, 0 ⩽ |a_{i}| ⩽ b_{i} ⩽ {(\frac{32^{1 / 2} ⌈l o g_{2} n⌉}{ε})}^{2 ⌈l o g_{2} n⌉}$ . One consequence...

A note on the Hausdorff-Besicovitch dimension of systems of linear forms

M. Dodson (1984)

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A function from Diophantine approximations.

Shiu, P. (1999)

Publications de l'Institut Mathématique. Nouvelle Série

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Counting rational points near planar curves

Ayla Gafni (2014)

Acta Arithmetica

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We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².