Approximation to Irrationals by Classes of Rational Numbers

if and only if k^l/51'2. Scott [ll] proved that if the fractions a/b are restricted to any one of the three classes (i) a, b both odd, (ii) a even, b odd, or (iii) a odd, b even, the same conclusion holds if &=g 1. Other proofs of this have been given by Oppenheim [8], Robinson [lO], and Kuipers and Meulenbeld [6]. Robinson also showed that if any pair of these classes were used, then &2il/2. Let (m, r, s) = 1; then the set of all fractions a/b in lowest terms for which a = r, b=s (mod m) will be denoted by (r, s). Descombes and Poitou [l; 9] have investigated the values of k needed for sets (r, s). Hartman [2] and Koksma [5] have considered the problem of a universal constant for all sets of fractions a/b with a=r, b = s (mod m) where a, b need not be relatively prime nor is it required that (r, s, m) = 1. We obtain results for other classes of rational numbers. Let the continued fraction expansion of £ be £ = [do, d\, • • • ]; then the nth convergent is a„/6„=[do, d\, • • • , dn] and the nth denominator is dn. We shall use the following known results [4; 8; 10].