Diophantine approximation

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.

Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods.

The 2022 Fields Medal was awarded to James Maynard, in part for his work on Diophantine approximation.

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.

[6] Equivalently, a number is badly approximable if and only if its Markov constant is finite and its simple continued fraction is bounded.

It may be remarked that the preceding proof uses a variant of the pigeonhole principle: a non-negative integer that is not 0 is not smaller than 1.

This apparently trivial remark is used in almost every proof of lower bounds for Diophantine approximations, even the most sophisticated ones.

This link between Diophantine approximations and transcendental number theory continues to the present day.

The main improvements are due to Axel Thue (1909), Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955), leading finally to the Thue–Siegel–Roth theorem: If x is an irrational algebraic number and ε > 0, then there exists a positive real number c(x, ε) such that holds for every integer p and q such that q > 0.

All preceding lower bounds are not effective, in the sense that the proofs do not provide any way to compute the constant implied in the statements.

This means that one cannot use the results or their proofs to obtain bounds on the size of solutions of related Diophantine equations.

Nevertheless, a refinement of Baker's theorem by Feldman provides an effective bound: if x is an algebraic number of degree n over the rational numbers, then there exist effectively computable constants c(x) > 0 and 0 < d(x) < n such that holds for all rational integers.

Adolf Hurwitz (1891)[7] strengthened this result, proving that for every irrational number α, there are infinitely many fractions

such that: So equivalence is defined by an integer Möbius transformation on the real numbers, or by a member of the Modular group

The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of Serret: Theorem: Two irrational numbers x and y are equivalent if and only if there exist two positive integers h and k such that the regular continued fraction representations of x and y satisfy for every non negative integer i.

[11] Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation.

-approximable if there exist infinitely many rational numbers p/q such that Aleksandr Khinchin proved in 1926 that if the series

In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture.

For this function, the relevant series converges and so Khinchin's theorem tells us that almost every point is not

The Jarník-Besicovitch theorem, due to V. Jarník and A. S. Besicovitch, states that the Hausdorff dimension of this set is equal to

For this function, the relevant series diverges and so Khinchin's theorem tells us that almost every number is

So an appropriate analogue of the Jarník-Besicovitch theorem should concern the Hausdorff dimension of the set of badly approximable numbers.

This result was improved by W. M. Schmidt, who showed that the set of badly approximable numbers is incompressible, meaning that if

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.

It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.

In his plenary address at the International Mathematical Congress in Kyoto (1990), Grigory Margulis outlined a broad program rooted in ergodic theory that allows one to prove number-theoretic results using the dynamical and ergodic properties of actions of subgroups of semisimple Lie groups.

The work of D. Kleinbock, G. Margulis and their collaborators demonstrated the power of this novel approach to classical problems in Diophantine approximation.

Various generalizations of the above results of Aleksandr Khinchin in metric Diophantine approximation have also been obtained within this framework.