On the exponent of convergence of the digit sequence of Engel series

摘要

For x is an element of (0, 1), let < d(1) (x), d(2) (x), d(3) (x), ...> be the Engel series expansion of x. Denote by lambda(x) the exponent of convergence of the sequence {d(n)(x)}, namely @@@ lambda(x) = inf {s >= 0 : Sigma (n >= 1) d(n)(-s) (x) < infinity}. @@@ It follows from Erdos, Renyi and Szusz (1958) that lambda(x) = 0 for Lebesgue almost all x is an element of (0, 1). This paper is concerned with the topological and fractal properties of the level set {x is an element of (0, 1) : lambda(x) = alpha} for alpha is an element of [0, infinity]. For the topological properties, it is proved that each level set is uncountable and dense in (0, 1), and the level set is of first category for alpha is an element of [0, infinity) but residual for alpha = infinity. For the fractal properties, we prove that the Hausdorff dimensions of the level sets are as follows: @@@ dim(H) {x is an element of (0, 1) : lambda (x) = alpha } = {1-alpha, 0 <= alpha <= 1; @@@ 0, 1 < alpha <= infinity.

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