ON THE APPROXIMATION OF IRRATIONALS BY RATIONALS

Let xi be a real irrational number, and a greater than or equal to 0, b greater than or equal to 0, s > 1 be integers. A theorem of S. UCHIY AMA states that there are infinitely many pairs of integers u and v no t equal 0 such that \xi- u/v\ < s(2)/4v(2) and u = a, v = b mod s, pro vided that it is not a = b = 0 mod s. It is shown that, this result is best - possible for all integers s > 1.