Talk:Diophantine approximation

Terminology

A question has been raised [1] about the terminolgy "partial quotient". In number theoretic texts such as Hardy & Wright, for a continued fraction [a0;a1,a2,a3,...] the ai are the "partial quotients", the pn/qn are the "convergents". Jones & Thron (1980) use different terminology for the generalised continued fraction b1,a1;b2,a2;,...] where the bi,ai are the "partial numerator" and "partial denominator" respectively. I suggest that "partial quotient" is the conventional terminology for number theoretic articles. Deltahedron (talk) 07:40, 18 November 2012 (UTC)

I have no personal opinion on this matter, but convergent (continued fraction) uses Jones & Thron terminology. As Hardy and Wright seems more notable (I have never heard before from Jones & Thron, but frequently from Hardy an Wright), I suggest to edit convergent (continued fraction). D.Lazard (talk) 08:35, 18 November 2012 (UTC)

The German terminology of Perron is a combination of both: here Teilzähler, Teilnenner and Teilbruch are used (meaning partial nominator, denominator and quotient). Apparently in English partial quotient is used for the denominator, which is fine if the nominators are 1. -- KurtSchwitters (talk) 14:16, 25 November 2012 (UTC)

Algorithm to Compute Rational Approximations

Would it be useful to include Euclid's algorithm, or a link to it (http://en.wikipedia.org/wiki/Euclid%27s_algorithm)? This computes rational approximations to any given real number. 132.244.72.5 (talk) 13:41, 26 November 2012 (UTC)

It is not Euclid's algorithm (which computes greatest common divisor) but Euclidean division that computes rational approximations of any real number. Euclidean division is also the tool that allows to compute the continued fraction expansion of any real number. As the second paragraph of the lede and a section refer to continued fraction for approximating a real number, the right place to mention Euclidean division seems to be in Continued fraction. Also, Euclidean division article should mention that an important application of Euclidean division is the computation of continued fraction expansions. D.Lazard (talk) 14:41, 26 November 2012 (UTC)

Mistake in Khinchin's Theorem

The section on Khinchin's Theorem previously read:

Aleksandr Khinchin proved in 1926 that if the series $\sum _{q}\psi (q)$ diverges, then almost every real number (in the sense of Lebesgue measure) is $\psi$-approximable, and if the series converges, then almost every real number is not $\psi$-approximable.

However, the last example in the paper by Duffin and Schaeffer proves that this is false as stated. Khinchin's Theorem originally included the assumption that $q\psi (q)$ is non-increasing, which I've now added to the article. — Preceding unsigned comment added by Ludosoph (talk • contribs) 19:00, 2 December 2017 (UTC)

Edit: Reverted the above change. The example by Duffin and Schaeffer only shows that $\psi$ needs to be non-increasing. The original result by Khinchin required $q\psi (q)$ to be non-increasing, but apparently Wolfgang M. Schmidt improved this to only require that $\psi (q)$ be non-increasing. Can anyone who is more familiar with this subject confirm or deny this? Ludosoph (talk) 16:58, 7 December 2017 (UTC)

Definition of \psi-approximable

In the definition of \psi-approximable, right-hand-side of the equation, why is the denominator the absolute value of q? If q is supposed to be positive, as I guess is assumed throughout the article, taking the absolute is unnecessary. If q is allowed to be negative, one would also have to use |q| in the argument of \psi, since the function is supposed to be defined only on positive integers.

Not a mistake, just slightly confusing, and aesthetically unsatisfying. The current notation looks a bit like a mix of different conventions. — Preceding unsigned comment added by 213.55.220.55 (talk) 22:58, 27 May 2020 (UTC)

Hurwitz Theorem

The article states that: Over the years, this theorem has been improved until the following theorem of Émile Borel (1903). For every irrational number α, there are infinitely many fractions ${\tfrac {p}{q}}\;$ such that

$\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}\,.$

I am not an expert, but I think this result is due to Hurwitz, 1891. At least, that's what this reference says: https://www.fmi.uni-sofia.bg/sites/default/files/biblio/fulltext/89-047-058.pdf

Borel, 1903, apparently strengthened this result to say that given three consecutive continued fraction convergents for α, at least one will satisfy the above inequality. At least, that's what the reference I posted says. And I believe that's what the cited theorem in Perron's book says, too, though I don't speak German.

Would someone who knows more about this than I do please take a look, and possibly fix it? Thanks!

Kier07 (talk) 21:45, 26 March 2022 (UTC)

OK, since no one seems to be rushing to correct the article, I'll do it myself. I will change it to make clear both Hurwitz and Borel's contributions. The Borel part is less essential, and could be removed if that's what people think is best. But we definitely should not be crediting him for Hurwitz's result. Kier07 (talk) 16:56, 29 March 2022 (UTC)