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

Abstract

Let B denote an infinite sequence of positive integers b1 < b2 < . . ., and let τ denote the exponent of convergence of the series ∞ n=1 1/bn; that is, τ = inf{s 0 :1] : an(x) ∈ B(n 1) and an(x) → ∞ as n → ∞}. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221-227] proved the inequality dimH 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.

Continued fractions with sequences of partial quotients over the field of Laurent series

“…In [16], Wang and the second author confirmed Hirst's conjecture without any assumption on Λ, that is,…”

A remark on continued fractions with sequences of partial quotients

“…In [1], Cusick proved Hirst's conjecture under some density assumption on B, he showed that if there exist constants c, q and r depending only on B such that r < τq and, for all real p q, the sequence B has at least cn τp−r members in every interval [(n − 1) p , n p ], then dim H E(B) = τ (B) 2 . In [7], we gave a positive answer to Hirst's conjecture without any assumption on B, that is…”

A generalization of the Jarník–Besicovitch theorem by continued fractions

“…The Hausdorff dimension of this set was well studied by Bugeaud [5,7] and Bugeaud and Moreira [8] by also using the tools of continued fractions. For more dimensional results relating the partial quotients, see [9,10,13,16,17,19,22,23,25,26,34,35,37] and references therein.…”