Talk:Diophantine approximation
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[edit] 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 [1], 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?
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- 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)
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- 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)
[edit] 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)
