Diophantine approximation

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In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers.

The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number a/b is a "good" approximation of a real number α if the absolute value of the difference between a/b and α may not decrease if a/b is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of 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 allowed Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that π and e are transcendental were obtained with a similar method.

Thus Diophantine approximations and transcendence theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations.

Best Diophantine approximations of a real number[edit]

Given a real number α, there are two ways to define a best Diophantine approximation of α. For the first definition,[1] the rational number p/q is a best Diophantine approximation of α if

\left|\alpha -\frac{p}{q}\right | < \left|\alpha -\frac{p'}{q'}\right |,

for every rational number p'/q' different from p/q such that 0 < q'q.

For the second definition,[2][3] the above inequality is replaced by

\left|q\alpha -p\right| < \left|q^\prime\alpha - p^\prime\right|.

A best approximation for the second definition is also a best approximation for the first one, but the converse is false.[4]

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.[3][4][5] For the first definition, one has to consider also the semiconvergents.[1]

For example, the constant e = 2.718281828459045235... has the (regular) continued fraction representation


Its best approximations for the second definition are

 3, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{87}{32}, \ldots\, ,

while, for the first definition, they are

3, \tfrac{5}{2}, \tfrac{8}{3}, \tfrac{11}{4}, \tfrac{19}{7}, \tfrac{30}{11},
\tfrac{49}{18}, \tfrac{68}{25}, \tfrac{87}{32}, \tfrac{106}{39}, \ldots\, .

Measure of the accuracy of approximations[edit]

The obvious measure of the accuracy of a Diophantine approximation of a real number α by a rational number p/q is \left|\alpha-\frac{p}{q}\right|. However, this quantity may always be made arbitrarily small by increasing the absolute values of p and q; thus 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 \left|\alpha-\frac{p}{q}\right|>\phi(q) ". In some case, "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 accounts 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 \left|\alpha-\frac{p}{q}\right|<\phi(q) ".

Badly approximable numbers[edit]

A badly approximable number is an x for which there is a positive constant c such that for all rational p/q we have

\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ .

The badly approximable numbers are precisely those with bounded partial quotients.[6]

Lower bounds for Diophantine approximations[edit]

Approximation of a rational by other rationals[edit]

A rational number \alpha =\frac{a}{b} may be obviously and perfectly approximated by \tfrac{p_i}{q_i} = \tfrac{i\,a}{i \,b} for every positive integer i.

If \tfrac{p}{q} \not= \alpha = \tfrac{a}{b}\,, we have

 \left|\frac{a}{b} - \frac{p}{q}\right| = \left|\frac{aq-bp}{bq}\right| \ge \frac{1}{bq},

because |aq-bp| is a positive integer and is thus not lower than 1. Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections).

It may be remarked that the preceding proof uses a variant of the pigeon hole 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.

In summary, a rational number is perfectly approximated by itself, but is badly approximated by any other rational number.

Approximation of algebraic numbers, Liouville's result[edit]

Main article: Liouville number

In the 1840s, Joseph Liouville obtained the first lower bound for the approximation of algebraic numbers: If x is an irrational algebraic number of degree n over the rational numbers, then there exists a constant c(x) > 0 such that

 \left| x- \frac{p}{q} \right| > \frac{c(x)}{q^{n}}

holds for every integers p and q where q > 0.

This result allowed him to produce the first proven example of a transcendental number, the Liouville constant

\sum_{j=1}^\infty 10^{-j!} = 0.110001000000000000000001000\ldots\,,

which does not satisfy Liouville's theorem, whichever degree n is chosen.

This link between Diophantine approximations and transcendence theory continues to the present-day. Many of the proof techniques are shared between the two areas.

Approximation of algebraic numbers, Thue–Siegel–Roth theorem[edit]

During more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound allows to prove that more numbers are transcendental. The main improvements are due to Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947) and Klaus Roth (1955), leading finally to the so-called Thue–Siegel–Roth theorem: If x is an irrational algebraic number and ε a (small) positive real number, then there exists a positive constant c(x, ε) such that

    \left| x- \frac{p}{q} \right|>\frac{c(x, \varepsilon)}{q^{2+\varepsilon}}

holds for every integers p and q such that q > 0.

In some sense, this result is optimal, as the theorem would be false with ε=0. This is an immediate consequence of the upper bounds described below.

Simultaneous approximations of algebraic numbers[edit]

Main article: Subspace theorem

Subsequently, Wolfgang M. Schmidt generalized this to the case of simultaneous approximations, proving that: If x1, ..., xn are algebraic numbers such that 1, x1, ..., xn are linearly independent over the rational numbers and ε is any given positive real number, then there are only finitely many rational n-tuples (p1/q, ..., pn/q) such that

|x_i-p_i/q|<q^{-(1+1/n+\varepsilon)},\quad i=1,\ldots,n.

Again, this result is optimal in the sense that one may not remove ε from the exponent.

Effective bounds[edit]

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. However, these techniques and results can often be used to bound the number of solutions of such 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

\left| x- \frac{p}{q} \right|>\frac{c(x)}{|q|^{d(x)}}

holds for all rational integers.

However, as for every effective version of Baker's theorem, the constants d and 1/c are so large that this effective result cannot be used in practice.

Upper bounds for Diophantine approximations[edit]

General upper bound[edit]

The first important result about upper bounds for Diophantine approximations is Dirichlet's approximation theorem, which implies that, for every irrational number α, there are infinitely many fractions \tfrac{p}{q}\; such that

\left|\alpha-\frac{p}{q}\right| < \frac{1}{q^2}\,.

This implies immediately that one can not suppress the ε in the statement of Thue-Siegel-Roth theorem.

Over the years, this theorem has been improved until the following theorem of Émile Borel (1903).[7] 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}\,.

Therefore \frac{1}{\sqrt{5}\, q^2} is an upper bound for the Diophantine approximations of any irrational number. The constant in this result may not be further improved without excluding some irrational numbers (see below).

Equivalent real numbers[edit]

Definition: Two real numbers x,y are called equivalent[8][9] if there are integers a,b,c,d\; with ad-bc = \pm 1\; such that:

y = \frac{ax+b}{cx+d}\, .

So equivalence is defined by an integer Möbius transformation on the real numbers, or by a member of the Modular group \text{SL}_2^{\pm}(\Z), the set of invertible 2 × 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an equivalence class for this relation.

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 there exist two positive integers h and k such that the regular continued fraction representations of x and y

x=[u_0; u_1, u_2, \ldots]\, ,
y=[v_0; v_1, v_2, \ldots]\, ,



for every non negative integer i.[10]

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

Lagrange spectrum[edit]

Main article: Markov spectrum

As said above, the constant in Borel's theorem may not improved, as shown by Adolf Hurwitz in 1891.[11] Let \phi = \tfrac{1+\sqrt{5}}{2} be the golden ratio. Then for any real constant c with c > \sqrt{5}\; there are only a finite number of rational numbers p/q such that

\left|\phi-\frac{p}{q}\right| < \frac{1}{c\, q^2}.

Hence an improvement can only be achieved, if the numbers which are equivalent to \phi are excluded. More precisely:[12][13] For every irrational number \alpha, which is not equivalent to \phi, there are infinite many fractions \tfrac{p}{q}\; such that

\left|\alpha-\frac{p}{q}\right| < \frac{1}{\sqrt{8} q^2}.

By successive exclusions — next one must exclude the numbers equivalent to \sqrt 2 — of more and more classes of equivalence, the lower bound can be further enlarged. The values which may be generated in this way are called Lagrange spectrum. They converge to the number 3 and are related to the Markov numbers.[14][15]

Khinchin's theorem and extensions[edit]

Aleksandr Khinchin proved in 1926 that if \phi is a non-increasing function from the positive integers to the positive real numbers such that \sum_{q} \phi(q) < \infty\,, then for almost all real numbers x (not necessarily algebraic), there are at most finitely many rational p/q and

\left| x- \frac{p}{q} \right| < \frac{\phi(q)}{|q|}.

Similarly, if the sum diverges, then for almost all real numbers, there are infinitely many such rational numbers p/q.[16][17][18]

In 1941, R. J. Duffin and A. C. Schaeffer[19] proved a more general theorem that implies Khinchin's result, and made a conjecture now known by their name as the Duffin–Schaeffer conjecture. In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result is published in the Annals of Mathematics.[20]

Uniform distribution[edit]

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 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.

Unsolved problems[edit]

There are still simply-stated unsolved problems remaining in Diophantine approximation, for example the Littlewood conjecture and the Lonely runner conjecture. It is also unknown if there are algebraic numbers with unbounded coefficients in their continued fraction expansion.

Recent developments[edit]

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. Among its notable successes are the proof of the decades-old Oppenheim conjecture by Margulis, with later extensions by Dani and Margulis and Eskin–Margulis–Mozes, and the proof of Baker and Sprindzhuk conjectures in the Diophantine approximations on manifolds by Kleinbock and Margulis. Various generalizations of the above results of Aleksandr Khinchin in metric Diophantine approximation have also been obtained within this framework.

See also[edit]


  1. ^ a b Khinchin 1997, p. 21
  2. ^ Cassels 1957, p. 2
  3. ^ a b Lang 1995, p. 9
  4. ^ a b Khinchin 1997, p. 24
  5. ^ Cassels 1957, pp. 5–8
  6. ^ Bugeaud 2012, p. 245
  7. ^ Perron 1913, Chapter 2, Theorem 15
  8. ^ Hurwitz 1891, p. 284
  9. ^ Hardy & Wright 1979, Chapter 10.11
  10. ^ See Perron 1929, Chapter 2, Theorem 23, p. 63
  11. ^ Hardy & Wright 1979, p. 164
  12. ^ Cassels 1957, p. 11
  13. ^ Hurwitz 1891
  14. ^ Cassels 1957, p. 18
  15. ^ See Michel Waldschmidt: Introduction to Diophantine methods irrationality and transcendence, pp 24–26.
  16. ^ Cassels 1957, p. 120
  17. ^ Schmidt 1996, p. 39
  18. ^ Khinchin 1997, pp. 69–71
  19. ^ Duffin, R. J.; Schaeffer, A. C. (1941). "Khintchine's problem in metric diophantine approximation". Duke Mathematical Journal 8: 243–255. doi:10.1215/s0012-7094-41-00818-9. ISSN 0012-7094. Zbl 0025.11002. 
  20. ^ Beresnevich, Victor; Velani, Sanju (2006). "A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures". Annals of Mathematics 164: 971–992. doi:10.4007/annals.2006.164.971. Zbl 1148.11033. 


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