On the Approximation of Irrationals by Rationals

Abstract

Let ξ be a real irrational number, and a ≧ 0, b ≧ 0, s > 1 be integers. A theorem of S. UCHIYAMA states that there are infinitely many pairs of integers u and v ≠ 0 such that OVBARξ−u/vOVBAR ≤ s2/4v2 and u a, v b mod s, provided that it is not a 6 0 mod s. It is shown that this result is best-possible for all integers s > 1.