2008/02/01 by Bao-Wei Wang, Baowei Wang, Jun Wu · 3 citations
Mathematics · #Analytic Number Theory Research #Mathematical functions and polynomials #Mathematics and Applications
paper · doi:10.1112/blms/bdm103
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 : ∑n = 1∞ 1/bns < ∞. Define E(B) = x ∈ [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.