Talk:Diophantine approximation

take the mathematical view

I have restructured the article , mainly the introduction. The section of A-S-R-Theorem was two-times inserted! We need no fighters here(!) to over-appreciated this! :-( Look were should it be and the development after (Kinchin, ...) begins? Any reason not to mention other famous mathematics who also contribute to this long-fought-for theorem?

• "The theory of continued fractions, as applied to square roots of integers and other quadratic irrationals, was studied from a Diophantine point-of-view by Fermat, Euler and others."

Better we have a reference here, also because here starts the Diophantine approximation historically. I saved it, in spite of this is not my work/formulation and I have not checked for historic facts about it. It must be early in the article, but I'm not sure whether to put it at the beginning of the section "basic Diophantine approximation".

• I wonder why this common definition of Notation for number sets is a good idea at the beginning?::* The size of the introduction in relation the the whole article should be balanced! In the stub/start version where I started a week ago, content of the body was put into the lead, because basic sections were missing in the body, resulting in such an ill-balanced overweighted introduction that times. E.g. the discussion how "approximation quality" should be formally defined and how it can't is no point for an introduction.

BTW: thanks to those people for doing grammar, typo, wording and stylish rewriting. Achim1999 (talk) 13:57, 20 June 2012 (UTC)

I wonder why this common definition of Notation for number sets is a good idea at the beginning?

I put it there because when I was reading the article I couldn't remember whether $\mathbb {N}$ included zero or not.

Virginia-American (talk) 14:46, 20 June 2012 (UTC)

Okay. Mathematicians can grasp this easily from the context, but not everybody is one. :) But this explains only a fourth of this "notation" section. ;) Maybe a better way in doing this is to use wiki-links on the first use of these(this) mathematical set-symbol(s)? (Like in other articles) Achim1999 (talk) 17:13, 20 June 2012 (UTC)

In fact the notation N for the positive integers is far from universal within mathematics; for example, within combinatorics typically N includes 0. It would be far better to switch to some notation like $\mathbf {Z} _{>0}$ and $\mathbf {Z} _{\geq 0}$ that is not inherently ambiguous. --Joel B. Lewis (talk) 20:13, 28 June 2012 (UTC)

We need no notation-agitator on wikipedia. Moreover the article belongs to number theory. Achim1999 (talk) 20:18, 28 June 2012 (UTC)

I have no idea what your first sentence means, and I can't see how to read it so that it might relate to the content of what I wrote. My comment has two parts: (1) the notation that you say that all mathematicians understand, and which is described in the article as standard, is not in fact standard; (2) it is better to use notations that are unambiguous. (I now take the opportunity to add that $\mathbb {N} _{0}$ is an awful notation, and I would seek to avoid it even if it weren't ambiguous.) What part of these statements do you disagree with?

Actually looking more closely at the article, I also suggest replacing most instances of formulas like $\alpha \in \mathbb {R}$ with words like "the real number α" -- they have the same content, and the latter form is more accessible for most readers. --Joel B. Lewis (talk) 20:38, 28 June 2012 (UTC)

A mistake?

In Section "Measure of the accuracy of approximations" I read: 'In some case, "every rational number" may be replaced by "every rational number but a finite number"'. A finite number?? Boris Tsirelson (talk) 13:53, 20 August 2012 (UTC)

Further, in Section "Approximation of algebraic numbers, Thue–Siegel–Roth theorem" I read: "In some sense, this result is optimal, as the theorem is false if ." Naturally, I wonder, if what? Boris Tsirelson (talk) 14:00, 20 August 2012 (UTC)

For me the wording "In some sense, this result is optimal, as the theorem is false if" is followed by "(Greek letter epsilon) = 0". In other words, if you put epsilon equal to zero you get a statement which is not true: I have edited it to make it clearer, I hope. Perhaps your browser is not rendering Greek letters properly? Deltahedron (talk) 07:38, 19 November 2012 (UTC)

In section "Lagrange spectrum" there are two error bounds. The first is abs(phi - p/q) is less than 1/(c. q^2) ... this seems OK. The second is abs(alpha - p/q) is less than 1/(root(8).p^2). I thought all these error bounds had q in the denominator. This one has p in the denominator. Should it be q instead? Wikipedia is like the well of knowledge. You are shown the well of knowlege and power and are told that on any day except Friday, you can ask a question and the answer will come out of the well. If you are bold enough to ask "Why not Friday?" then a voice will come out of the well - "Because on Friday it's your turn in the well." After pausing to absorb this amusing little tale, I ask, is this not the human race in microcosm? When we have difficult questions to answer, we don't appeal to a higher power. All we can do is ask each other. How exactly do we ask each other? There are many ways and one of them is wikipedia. (188.220.70.97 (talk) 23:06, 18 November 2012 (UTC))

Yes, that p was a mistake. Corrected. Deltahedron (talk) 07:38, 19 November 2012 (UTC)

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)