Rational Approximations to Irrational Numbers

Abstract

A classical theorem of A. Hurwitz concerning rational approximations to irrational numbers states that for any real irrational number $\xi$ there are infinitely many pairs of integers $p,$ $q$ with $q>0$ satisfying the inequality (1)where the constant $\sqrt{5}$ on the right-hand side of (1) is the best possible number.S. Hartman [1], imposing congruential conditions on the denominator and the numerator of approximating fractions, has proved that if $\xi$ is an irrational number and if $a,$ $b$ and $s$ are fixed integers with $s>0$, then there exists an infinity of pairs of integers $u,$ $v$ with $v>0$ satisfying the conditions (2) and(3) $u\equiv a(mod s)$ , $v\equiv b(mod s)$ .As is noticed by Hartman, the exponent 2 of $s$ in the right side of (2) cannot be reduced in general; this will readily be seen by putting $a=b=0$.This result of Hartman has been improved in two ways by J. F. Koksma[2], who proved that if $\xi$ is an irrational number, then for any fixed real number $\epsilon>0$ and for any integers $s>0,$ $a,$ $b$ there exist infinitely many pairs of integers $u,$ $v$ with $v>0$ such that (4)$|\xi-\frac{u}{v}|<\frac{(1+\epsilon}{\sqrt{5}}\frac{)s^{2}}{v^{2}}$ , $u\equiv a(mod s)$ , $v\equiv b(mod s)$ , and that if $\xi$ is an irrational number and if $s>0,$ $a,$ $b$ are integers such that we have not simultaneously $a\equiv 0(mod s)$ and $b\equiv 0(mod s)$ , then there are infinitely many pairs of integers $u,$ $v$ with $v\neq 0$ satisfying

Locations

• Mathematics of Computation - View
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
• Terrestrial Environment Research Center (University of Tsukuba) - View - PDF