The Littlewood conjecture states that for any two real numbers ,where denotes the nearest integer function.In layman's terms, this conjecture concerns the simultaneous approximation of two real numbers by rationals, indeed saying that any two real numbers and can be simultaneously approximated at least moderately well by rationals having the same denominator (Venkatesh 2007).Though proof of the Littlewood conjecture still remains an open problem, many partial results exist. For example, Borel showed that the set of exceptional pairs of real numbers and for which the conjecture fails has Lebesgue measure zero. Much later, Einsiedler et al. (2006) proved that the set of pairs of exceptional points also has Hausdorff dimension zero.
For any algebraic number of degree , a rational approximation to must satisfyfor sufficiently large . Writing leads to the definition of the irrationality measure of a given number. Apostol (1997) states the theorem in the slightly modified but equivalent form that there exists a positive constant depending only on such that for all integers and with ,
If is a given irrational number, then the sequence of numbers , where , is dense in the unit interval. Explicitly, given any , , and given any , there exists a positive integer such thatTherefore, if , it follows that . The restriction on can be removed as follows. Given any real , any irrational , and any , there exist integers and with such that
As Lagrange showed, any irrational number has an infinity of rational approximations which satisfy(1)Furthermore, if there are no integers with and (corresponding to values of associated with the golden ratio through their continued fractions), then(2)and if values of associated with the silver ratio are also excluded, then(3)In general, even tighter bounds of the form(4)can be obtained for the best rational approximation possible for an arbitrary irrational number , where the are called Lagrange numbers and get steadily larger for each "bad" set of irrational numbers which is excluded.
The end of the last gap in the Lagrange spectrum,given by(OEIS A118472).Real numbers greater than are members of the Markov spectrum.
Given any real number and any positive integer , there exist integers and with such thatA slightly weaker form of the theorem states that for every real , there exist integers and with and such that
Every irrational number has an approximation constant defined bywhere is the nearest integer to and is the infimum limit. The quantity measures how well is approximated by the rational number . is said to be badly approximable if . An irrational number is badly approximable iff the terms of its continued fraction are bounded. Since quadratic surds have periodic continued fractions, they are badly approximable. The maximum possible value of is , attained for example at , where is the golden ratio.
For any real number , an irrational number can be approximated by infinitely many rational fractions in such a way thatIf , this becomes Hurwitz's irrational number theorem.
For algebraic with , has finitely many solutions. Klaus Roth received a Fields medal for this result.
If is any number and and are integers, then there is a rational number for which(1)If is irrational and is any whole number, there is a fraction with and for which(2)Furthermore, there are an infinite number of fractions for which(3)(Hilbert and Cohn-Vossen 1999, pp. 40-44).Hurwitz has shown that for an irrational number (4)there are infinitely rational numbers if , but if , there are some for which this approximation holds for only finitely many .
There are at least two Siegel's theorems. The first states that an elliptic curve can have only a finite number of points with integer coordinates.The second states that if is an algebraic number of degree , then there is an depending only on such thatfor all integer and (Landau 1970, pp. 37-56; Hardy 1999, p. 79).