List questions from diophantine-approximation

Published on 02 Oct 2022 - 6:35

91

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Given $\alpha\in \mathbb{R},$ do there exist arbitrarily large $m,k$ such that $\vert \alpha - \frac{k}{m}\vert < \frac{1}{m^2}$?

Published on 23 Feb 2026 - 6:03

103

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Diophantine approximation theory and "logarithmic" rather than "linear" error

Published on 03 Oct 2022 - 3:49

58

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Almost Diophantine approximation

Published on 04 Oct 2022 - 20:46

77

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Measuring how good the best possible rational approximation of a real number is

Published on 09 Oct 2022 - 0:58

236

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Reversing the coefficients of the continued fraction expansion of a rational number

Published on 15 Oct 2022 - 8:52

326

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Are there any perfect squares of the form $\underbrace{88\cdots8}_{n\text{ times}}1$ (in decimal, at least two $8$'s)?

Published on 27 Oct 2022 - 0:28

48

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Rational approximations of logarithms of ratios of primes

Published on 30 Oct 2022 - 7:18

63

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Absolute convergence of $\sum_{n=1}^\infty n^{-5} \csc(2n\pi \sqrt[3]{3})$

Published on 30 Nov 2022 - 21:31

159

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Do three arithmetic sequences of real numbers that include $0$ always get arbitrarily close to each other at nonzero numbers?

Published on 13 Dec 2022 - 2:00

164

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Can we scale the elements of a set of real numbers so that they are arbitrarily close to integers?

Published on 23 Dec 2022 - 3:36

24

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Does a limit exist for the fraction of $p\alpha_i$?

Published on 28 Dec 2022 - 14:15

196

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How is Siegel's lemma applied in number theory?

Published on 29 Dec 2022 - 12:26

126

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Equality case in Dirichlet's approximation theorem

Published on 01 Jan 2023 - 9:46

List Question

How well approximated by rationals are almost all real numbers?

Given $\alpha\in \mathbb{R},$ do there exist arbitrarily large $m,k$ such that $\vert \alpha - \frac{k}{m}\vert < \frac{1}{m^2}$?

Diophantine approximation theory and "logarithmic" rather than "linear" error

Almost Diophantine approximation

Measuring how good the best possible rational approximation of a real number is

Reversing the coefficients of the continued fraction expansion of a rational number

Are there any perfect squares of the form $\underbrace{88\cdots8}_{n\text{ times}}1$ (in decimal, at least two $8$'s)?

Rational approximations of logarithms of ratios of primes

Absolute convergence of $\sum_{n=1}^\infty n^{-5} \csc(2n\pi \sqrt[3]{3})$

Do three arithmetic sequences of real numbers that include $0$ always get arbitrarily close to each other at nonzero numbers?

Can we scale the elements of a set of real numbers so that they are arbitrarily close to integers?

Does a limit exist for the fraction of $p\alpha_i$?

How is Siegel's lemma applied in number theory?

Equality case in Dirichlet's approximation theorem

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