GENERALIZED CANTOR-INTEGERS AND INTERVAL DENSITY OF HOMOGENEOUS CANTOR SETS

Abstract

In this paper, we study the generalized Cantor-integers [Formula: see text] with the base conversion function [Formula: see text] being strictly increasing and satisfying [Formula: see text] and [Formula: see text]. We show that the sequence [Formula: see text] with [Formula: see text] is dense in the closed interval with the endpoints being its inferior and superior, respectively. Moreover, every homogeneous Cantor set [Formula: see text] satisfying open set condition can be induced by some generalized Cantor-integers, we get the exact point which attains the maximal interval density of the form [Formula: see text] with respect to the self-similar probability measure supported on [Formula: see text]. This result partially confirms a conjecture of E. Ayer and R. S. Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Am. Math. Soc. 351(9) (1999) 3725–3741].