A remark on continued fractions with sequences of partial quotients

Abstract

Given any infinite set B of positive integers b 1 < b 2 < ⋯ , let τ ( B ) denote the exponent of convergence of the series ∑ n = 1 ∞ b n − s . Let E ( B ) be the set { x ∈ [ 0 , 1 ] : a n ( x ) ∈ B ( n ⩾ 1 ) and a n ( x ) → ∞ as n → ∞ } . Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973) 221–227] proved the inequality dim H E ( B ) ⩽ τ ( B ) 2 and conjectured (see Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973), p. 225] and Cusick [T.W. Cusick, Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford Ser. (2) 41 (1990), p. 278]) that equality holds in general. In [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press], we gave a positive answer to this conjecture. In this note, we further show that the result in [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press] is sharp.