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

Abstract

Let B denote an infinite sequence of positive integers b 1 < b 2 < …, and let τ denote the exponent of convergence of the series ∑ n = 1 ∞ 1/b n ; that is, τ = inf {s ≥ 0 : ∑ n = 1 ∞ 1/b n s < ∞}. Define E(B) = {x ∈ [0, 1]: a n (x) ∈ B (n ≥ 1) and a n (x) → ∞ as n → ∞}. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221-227] proved the inequality dim H E(B) ≤ τ/2 and conjectured (see ibid., p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990) p. 278]) that equality holds. In this paper, we give a positive answer to this conjecture.