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Let $ξ = [a_{0}; a_{1}, a_{2}, \dots, a_{i}, \dots]$ be an irrational number in simple continued fraction expansion, $p_{i} / q_{i} = [a_{0}; a_{1}, a_{2}, \dots, a_{i}]$ , $M_{i} = q_{i}^{2} | ξ - p_{i} / q_{i} |$ . In this note we find a function $G (R, r)$ such that $\begin{matrix} M_{n + 1} < R and M_{n - 1} < r imply M_{n} > G (R, r), & M_{n + 1} > R and M_{n - 1} > r imply M_{n} < G (R, r) . \end{matrix}$ Together with a result the author obtained, this shows that to find two best approximation functions $\tilde{H} (R, r)$ and $\tilde{L} (R, r)$ is a well-posed problem. This problem has not been solved yet.

The Conjugate Property of the Borel Theorem on Diophantine Approximation.

Jingcheng Tong (1983)

Mathematische Zeitschrift

The Connection between the Zeros of the ...-Function and Sequences (g(p)), p Prime, mod 1.

Johannes Schoißengeier (1979)

Monatshefte für Mathematik

The continued fraction expansion of α with μ(α) = 3

Shin-Ichi Yasutomi (1998)

Acta Arithmetica

The converse to Alexandre Froda's irrationality criterion.

David Sanker (1991)

Mathematica Scandinavica

The critical determinant of the double paraboloid and Diophantine approximation in $ℝ^{3}$ and $ℝ^{4}$ .

Nowak, Werner Georg (1999)

Mathematica Pannonica

The $d$ -dimensional Gauss transformation: Strong convergence and Lyapunov exponents.

Hardcastle, D.M., Khanin, K. (2002)

Experimental Mathematics

The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves

Yasutsugu Fujita (2007)

Acta Arithmetica

The density of rational and integral points on algebraic varieties

M. L. Brown (1992)

Annales scientifiques de l'École Normale Supérieure

The digamma function, Euler-Lehmer constants and their p-adic counterparts

T. Chatterjee, S. Gun (2014)

Acta Arithmetica

The goal of this article is twofold. First, we extend a result of Murty and Saradha (2007) related to the digamma function at rational arguments. Further, we extend another result of the same authors (2008) about the nature of p-adic Euler-Lehmer constants.

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The algebraic groups leading to the Roth inequalities

The algebraic independence of certain numbers to algebraic powers

The algebraic independence of Weierstrass functions and some related numbers

The application of a new mean value theorem to the fractional parts of polynomials

The Approximation of Solutions to Linear Homogeneous Differential Equations by Rational Functions.

The Behavior of Rigid Analytic Functions Around Orbits of Elements of < $> ℂ_{p} <$ >

The best Diophantine approximation functions by continued fractions

The Conjugate Property of the Borel Theorem on Diophantine Approximation.

The Connection between the Zeros of the ...-Function and Sequences (g(p)), p Prime, mod 1.

The continued fraction expansion of α with μ(α) = 3

The converse to Alexandre Froda's irrationality criterion.

The critical determinant of the double paraboloid and Diophantine approximation in $ℝ^{3}$ and $ℝ^{4}$ .

The $d$ -dimensional Gauss transformation: Strong convergence and Lyapunov exponents.

The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves

The density of rational and integral points on algebraic varieties

The digamma function, Euler-Lehmer constants and their p-adic counterparts

The dimensions of integral points and holomorphic curves on the complements of hyperplanes

The diophantine equation $x^{2} = p^{a} \pm p^{b} + 1$

The discrepancy of (n...)n ... 1.

The Distribution of Farey Points.

Currently displaying 1 – 20 of 131