Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche

@article{HurwitzUeberDA,
  title={Ueber die angen{\"a}herte Darstellung der Irrationalzahlen durch rationale Br{\"u}che},
  author={Adolf Hurwitz},
  journal={Mathematische Annalen},
  volume={39},
  pages={279-284}
}
  • A. Hurwitz
  • Published 1 June 1891
  • Mathematics
  • Mathematische Annalen
Man kann bekanntlich jede Irrationalzahl a durch eine unbegrenzte Reihe von rationalen Bruchen $$\frac{{{x_1}}}{{{y_1}}},\;\frac{{{x_2}}}{{{y_2}}},\;\frac{{{x_3}}}{{{y_3}}},\; \ldots $$ (1) derart annahern, dass, abgesehen vom Vorzeichen, $$\alpha- \frac{{{x_n}}}{{{y_n}}}< \frac{1}{{y_n^2}}\quad \left( {n = 1,\;2,\;3 \ldots } \right)$$ (2) ist. Dieser Satz ergibt sich unmittelbar aus der Lehre von den Kettenbruchen; er lasst sich aber auch durch andere sehr… 

Directional Poincaré inequalities along mixing flows

We provide a refinement of the Poincaré inequality on the torus Td$\mathbb{T}^{d}$: there exists a set B⊂Td$\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every α∈B$\alpha \in

Roots of Markoff quadratic forms as strongly badly approximable numbers

For a real number $x$, $\| x\| = \min \{|x-p|: p\in Z\}$ is the distance of $x$ to the nearest integer. We say that two real numbers $\theta$, $\theta'$ are $\pm$ equivalent if their sum or

DIOPHANTINE APPROXIMATION IN $\mathbf{Q}(\sqrt{-5})$ AND $\mathbf{Q}(\sqrt{-6})$

The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the

Linear Forms in Logarithms

Hilbert’s problems form a list of twenty-three problems in mathematics published by David Hilbert, a German mathematician, in 1900. The problems were all unsolved at the time and several of them were

Approximation to irrationals by classes of rational numbers

if and only if k ? 1/5112. Scott [11] proved that if the fractions a/b are restricted to any one of the three classes (i) a, b both odd, (ii) a even, b odd, or (iii) a odd, b even, the same

Obfuscation of Big Subsets and Small Supersets as well as Conjunctions

TLDR
This work obfuscates the big subset and small superset functionalities in a very simple way and gives a proof of input-hiding for the conjunction obfuscation by Bartusek et al. (see Appendix A).

Small Superset and Big Subset Obfuscation

TLDR
This paper obfuscates SSF and BSF in a very simple and efficient way and proves both input-hiding security and virtual black-box (VBB) security based on the subset product problem.

Big Subset and Small Superset Obfuscation

TLDR
This paper obfuscates BSF and SSF in a very simple and efficient way, and proves both virtual black-box (VBB) security and input-hiding security in the standard model based on the subset product problem.
...