Some Arithmetical Properties of the Convergents of a Continued Fraction

where the al , a2 , . . . are positive integers . We show in § 1 that, for "almost all" ~, G(B,,) increases rapidly with n (Theorem 1) . In § 2, we prove that ~ is a Liouville number (i .e . B,, < Bell-1-1 for arbitrary E > 0 and an infinity of 7a) if G(B„) is bounded for all n (Theorem 2) ; and, in fact, there are Liouville numbers with bounded G(B„) . If the denominators a,,+1 are bounded or increase slowly, then -we can prove sharper results (B and C) ; but we omit the proofs, since they are similar to that of Theorem 2 . Corresponding results hold for the numerators A,, of the convergents A„ /8,,, of ~ .