Functiones et Approximatio Commentarii Mathematici

Counting Diophantine Approximations

Jörg Brüdern

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A recent development of the Davenport-Heilbronn method for diophantine inequalities is reexamined, and then applied to a class of problems in diophantine approximation. Among other things, an asymptotic formula is obtained for the number of solutions of the simultaneous inequalities $|n_j - \lambda_j n_0| <\varepsilon$ with square-free $n_j \in [1,N]$, whenever the positive real numbers $\lambda_1, \ldots, \lambda_r$ and $1$ are linearly independent over the rationals.

Article information

Funct. Approx. Comment. Math., Volume 39, Number 2 (2008), 237-260.

First available in Project Euclid: 19 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11J13: Simultaneous homogeneous approximation, linear forms
Secondary: 11D75: Diophantine inequalities [See also 11J25]

Davenport-Heilbronn method diophantine approximation square-free numbers


Brüdern, Jörg. Counting Diophantine Approximations. Funct. Approx. Comment. Math. 39 (2008), no. 2, 237--260. doi:10.7169/facm/1229696574.

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