Diophantine approximation (nonfiction)
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number p/q is a "good" approximation of a real number α if the absolute value of the difference between p/q and α may not decrease if p/q is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions.
Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than the lower bound for algebraic numbers, which is itself larger than the lower bound for all real numbers. Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number.
This knowledge enabled Joseph Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that π and e are transcendental were obtained by a similar method.
Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations.
The 2022 Fields Medal was awarded to James Maynard, in part for his work on Diophantine approximation.
Best Diophantine approximations of a real number
Given a real number α, there are two ways to define a best Diophantine approximation of α ...
The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. For the first definition, one has to consider also the semiconvergents.
Measure of the accuracy of approximations
the accuracy of the approximation is usually estimated by comparing this quantity to some function φ of the denominator q, typically a negative power of it.
For such a comparison, one may want upper bounds or lower bounds of the accuracy. A lower bound is typically described by a theorem like "for every element α of some subset of the real numbers and every rational number p/q, we have ... In some cases, "every rational number" may be replaced by "all rational numbers except a finite number of them", which amounts to multiplying φ by some constant depending on α.
For upper bounds, one has to take into account that not all the "best" Diophantine approximations provided by the convergents may have the desired accuracy. Therefore, the theorems take the form "for every element α of some subset of the real numbers, there are infinitely many rational numbers p/q such that ...
Uniform distribution
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 ... 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 throughout analytic number theory in the bounding of error terms.
Related to uniform distribution is the topic of irregularities of distribution, which is of a combinatorial nature.
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- Diophantine approximation @ Wikipedia
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- [ Post] @ Twitter (31 December 2025)
