Metrical Theory for Optimal Continued Fractions

where Ed= fl, bkEZ2,, k 2 1, and with some constraints on bk and sk. Usually we will assume that x is irrational, and thus that the expansion (1.1) is infinite. A special case of an SRCF is the regular continuedfraction, RCF, which is obtained by taking ck = 1 for every k in (1.1). The aim in introducing the OCF was to optimize two things simultaneously. In the first place one wishes the convergents pk/qk = Cbo; ~161, &, . . . . Ekbk], yielded by the SRCF, to be good approximations of x for every k 2 1, in the sense described below. For the regular continued fraction expansion RCF(x) = [B,; B,, B,, . ..] it is known that the convergents, denoted now by P,/Q, (n > l), all satisfy