A problem of Hirst on continued fractions with sequences of partial quotients

Let B denote an infinite sequence of positive integers b(1) < b(2) < ..., and let tau denote the exponent of convergence of the series (infinity)Sigma(n=1)1/b(n); that is, tau = inf {s >= 0 : 1/b(n)(s) < infinity}. Define E(B) {x is an element of [0,1] :a(n)(x) is an element of B(n >= 1) and a(n) (x) --> infinity as n --> infinity}. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221-2271 proved the inequality dim(H) E(B) <= tau/2 and conjectured (see ibid., p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990) p. 2781) that equality holds. In this paper, we give a positive answer to this conjecture.