| Diophantine Approximation |
Article Index for Diophantine |
Website Links For Diophantine |
Information AboutDiophantine Approximation |
| CATEGORIES ABOUT DIOPHANTINE APPROXIMATION | |
| number theory | |
| diophantine approximationnumber theory | |
| diophantine approximation | |
| number theory | |
|
The subject might be considered to be founded by the result of Liouville on general Algebraic Number s (the Lemma on the page for Liouville Number ). Before that, much was known from the theory of Continued Fraction s, as applied to square roots of integers and other quadratic irrationals. This result was improved by , a disadvantage in applications. Another topic that has seen a thorough development is the theory of Uniform Distribution Mod 1 . Take a sequence ''a''1, ''a''2, ... 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 approximation 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. 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 . |
|
|