Talk:Diophantine approximation/Archive 1

What does it mean?
Hi, as someone that does not already understand this subject thoroughly I thought I would be able to give some useful constructive advice to make the article more readable. I've copied the article text and would like to comment on a number of sections:

In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers. The smallness of the distance from the real number to be approximated and the rational number that approximates it is a crude measure of how good the approximation is.
 * What sense of measure makes sense here?
 * The second sentence is a bit clumsy; in this case absolute value of the difference is the metric used. Charles Matthews

A subtler measure considers how good the approximation is by comparison to the size of the denominator.
 * How is this subtler, and what comparison to the size of the denominator is made? Is smaller or larger better, and vs what?
 * If you use denominator 100, you can be more accurate than with denominator 10. Charles Matthews

The subject might be considered to be founded by the result of Liouville on general algebraic numbers (the Lemma on the page for Liouville number). Before that much was known from the theory of continued fractions, as applied to square roots of integers and other quadratic irrationals.
 * at this point, if "this result" is going to be referenced, a summarization of "the result" would be critical to the flow of understanding of the current article.

This result was improved by Axel Thue and others, leading in the end to a definitive theorem of Roth: the exponent in the theorem was reduced from n, the degree of the
 * Theorem of Roth? what is that? Ok I see this, but I have no idea what impact the finiteness of the solutions to that has, and what a solution to that even means. (That's clearly separate from this article, of course). A description and summary of Roth's theorem would be critical again to the understanding of what exponent is being referred to above.
 * We don't yet have the Thue-Siegel-Roth theorem article that would go into that; this result is one of the deepest in number theory. Charles Matthews

algebraic number, to any number greater than 2 (i.e. '2+epsilon'). After that generalisation was made to simultaneous approximation, by Schmidt. The proofs were difficult, and not effective, a disadvantage in applications.

Another topic that has seen a thorough development is the theory of uniform distribution mod 1. Take a sequence a1, a2, ... of real numbers and consider their fractional parts. That is, more abstractly, look at the sequence in R/Z, which is a circle. For any interval I on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer N, and compare it to the proportion of the circumference occupied by I. Uniform distribution means that in the limit, as N grows, the proportion of hits on the interval tends to the 'expected' value. Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine appproximation results were closely related to the general problem of cancellation in exponential sums, which occurs all over analytic number theory in the bounding of error terms.
 * This part just has me lost due to my lack of formal theoretical math training. Maybe it should be obvious why "the sequence in R/Z" is a circle, but what does R/Z even mean?
 * It's a quotient group; but imagine the unit interval [0,1]] bent round into a loop so that 0 = 1, and you get the picture. Charles Matthews

After Roth's theorem, the major advances in the subject have been in connection with transcendence theory. Related to uniform distribution is the topic of irregularities of distribution, which is of a combinatorial nature. There are still simply-stated unsolved problems remaining in Diophantine approximation, for example Littlewood's conjecture.

See also: low-discrepancy sequence.


 * I am familiar with those, but what connection does it have to Diophantine approximation?


 * Well, a great deal, since LD means equidistribution that is, in quantitative terms, very good. Charles Matthews 06:50, 11 Jun 2004 (UTC)


 * Unfortunately after reading the article I'm not entirely sure what the subject is, or if its applications are important. Hopefully my comments can help improve the article.  Who knows.  - Taxman 00:44, Jun 11, 2004 (UTC)


 * It's number theory - which happens to have applications, but is certainly not always developed for the sake of them. Charles Matthews 06:50, 11 Jun 2004 (UTC)

Continued fractions
I've managed to borrow a copy of Continued Fractions: Analytic Theory and Applications by Jones and Thron. Although the book is primarily concerned with the application of cf's to problems in complex analysis, the first chapter discusses the early history of the arithmetic theory of regular continued fractions.

Anyway, these authors trace the "best approximation" characteristic of regular cf's back to Daniel Schwenter (1625) and to a posthumously published result of C. Huygens (d. 1695). They also point out that "transcendental numbers" can be constructed (in light of Liouville's result about algebraic numbers) as regular continued fractions, and that these can be proven not to be algebraic of any degree n (i.e., transcendental) by application of an inequality discovered by Lagrange.

Anyway, the point is that the story of Diophantine approximation in its modern form really starts in the seventeenth century and not in the nineteenth century. I'll try to get some of this stuff written up and into the article within the next week or so. DavidCBryant 17:39, 22 May 2007 (UTC)

Lagrange's approximation theorem
Lagrange's approximation theorem redirects to this article, but it isn't mentioned anywhere here. If appropriate for this article: can somebody point out what this theorem is about? If not, maybe the theorem should get its own page (maybe a stub first, but a redirect to this page and no mention of the theorem is pretty misleading.) 134.169.77.186 (talk) 10:12, 6 January 2009 (UTC) (ezander)


 * I suspect, from the context at golden ratio, that it means the case of Dirichlet's approximation theorem that specialises &alpha; to a quadratic irrational. If so, the redirect (from 2005) should certainly go there instead. Charles Matthews (talk) 13:07, 6 January 2009 (UTC)

don't talk and die here, but build up the article (and then die) :-)
I have laid a hopefully good/improvable foundation of the most important and missing part. Now feel free to add your part to the article (of hopefully mathematical content). Achim1999 (talk) 17:04, 12 June 2012 (UTC)
 * We now have left the stub-level indisputable for this article, IMHO. :) Achim1999 (talk) 22:49, 12 June 2012 (UTC)
 * I have compared the present version of the article with the one preceding Achim1999 edits here are some comments
 * The new version should be tagged as too technical: after the lead, it starts directly into purely technical matters without any indication of the context. There is no one section for which the context is given.
 * The most important result related to the subject is that algebraic numbers can not have good approximation. In the previous version, this was not clearly stated, but this could easily be deduced from the lead and the first section. In Achim1999's version this fundamental property is very difficult to find.
 * Most of the first section (Basic diophantine approximation) should be replaced by a link to continuous function continuous fraction (sorry for the typo)
 * The new version lists the main results of the subject but never gives any indication on the reason for which they are useful and on their importance (this is almost the same thing)
 * In conclusion, Achim's version looks more like a review paper for people accustomed to the subject than like a encyclopedic article. I have not a clear idea of what to do. One may
 * Revert to the version before Achim1999's edits and then use Achim1999's material to expand this version into a correct article.
 * Reinsert the first section of the old version (it was pretty good) a new first section and do the minimal edits needed for a reasonable coherence
 * Other ideas?
 * For the moment, I prefer the second solution. D.Lazard (talk) 14:21, 15 June 2012 (UTC)


 * Finally I have rewritten the lead to emphasize Thue-Siegel-Roth theorem and I have restored the main part of the older version

as a new first section. D.Lazard (talk) 10:40, 16 June 2012 (UTC)


 * First of all: if you don't know this matter, don't change it to the worse. This is mathematic not a talk at your wellness! To be more specific:
 * What is "Diophantine_approximation" ? The stub before totally miss this issue! It looks like a collection of theorems and other stuff arbitrary put after each other. First take a introduction book for "Diophantine approximation" or a number theory book with a short overview to this topic. :-(
 * If you have read this, then you might be able to write a useful lead!
 * "::* Most of the first section (Basic Diophantine approximation) should be replaced by a link to continuous function" -- my advice: see the point above. Continued fractions are/was a tool to solve the older problem of "Diophantine approximation".
 * "I have not a clear idea of what to do" Yes, because you are missing the scientific background for this topic -- not to mention an overview. :-(
 * This article is not a wider view of Roth-Siegel-Theorem! And what you call "
 * I cite: "In modern mathematics, the main interest of the study of Diophantine approximations lies ..." this is your person judgment! Did you always start with "modern mathematics an introduction" ? Probably you because you don't know the old(er) foundations and results? And "the main interest of ..." is your person preferences/judgment/weight. Such a subjective statement has NOT to be in any encyclopedia! :-( Moreover what your phrase "good approximation" means, stays a secret.
 * If you miss most of the foundations in "Diophantine_approximation" put your hands away from a structure of this article! You may correct/typos/grammar/style -- this is a major miss of mine. :-(
 * And to learn more, you can also look for historic dates/developments, before starting the leadi calling the Thue–Siegel–Roth theorem. As you also said later "Diophantine approximation" started far before Liouville in 1844 with Fermat, Euler, etc.!!
 * "The most important result related to the subject ..." Ups! Why do you have the right to make this extremal judgment? :-(
 * I'm really frustrated/worried about what yesterday has happened to the article. :-( Achim1999 (talk) 13:02, 20 June 2012 (UTC)


 * To Achim1999: Please read carefully the policies wp:civility and wp:no personal attack. I can not accept your last post which breaks these policies. Maybe, I'll answer later to the content of your post, when I will be able to skip your personal attacks. D.Lazard (talk) 14:55, 20 June 2012 (UTC)
 * Please, what do You call a personal attack? I did not attack here anybody personally! Each contradiction was given with reasons based on topic-content and mathematical background knowledge! You should change your personal interpretation when reading constructive (content-based!) critics, if you are the one who is addressed by this critic. :-/ Achim1999 (talk) 17:25, 20 June 2012 (UTC)


 * After my answer, you have added to your

"The most important result related to the subject ..." Ups! Why do you have the right to make this extremal judgment? :-(
 * from your post. This is a personal attack as defined in D.Lazard (talk) 18:33, 20 June 2012 (UTC)
 * Then use other definitions! I'm REALLY SURE that you have insufficient knowledge of this topic to act as a reliable reviewer! You delete things YOU don't know independent of their importance, thus you can't judge correctly (and I believe you know it!) or present them right. Point to this sentence and the whole talk regarding your activities since I tried to build up the article to the Math-Portal, please! People like you prevent to reach certain higher quality levels on wikipedia! Sorry to say this frankly without using the diplomatic tone. :-( Achim1999 (talk) 19:32, 28 June 2012 (UTC)
 * Please parse D. Lazard at Zentralblatt and then revise your statement on insufficient knowledge of algebra and algebraic number theory.--LutzL (talk) 10:44, 29 June 2012 (UTC)

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 &alpha;" -- 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 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)