Irregular Cantor sets and singular functions associated with the decimal expansions of e and π

Source abstract

In this paper, we introduce a family of perfect, nowhere-dense subsets of [Formula: see text] that are not self-similar on any non-empty open set. These sets, denoted by [Formula: see text] and [Formula: see text], are defined by forbidding, at each decimal place, the digit that appears in the decimal expansion of the transcendental numbers [Formula: see text] and [Formula: see text]. Despite the lack of self-similarity, they are homogeneous Moran sets and share many metric properties with the classical Cantor set: zero Lebesgue measure, Hausdorff dimension [Formula: see text], and cardinality of the continuum. For each irregular Cantor set we construct an associated singular continuous monotone function — the irregular Cantor–Lebesgue function — which maps the Cantor set onto [Formula: see text] and is constant on every complementary interval. We prove that these functions are Hölder continuous with exponent [Formula: see text], nowhere differentiable on the Cantor set, and do not satisfy Banach’s condition [Formula: see text]. Their local convexity/concavity properties are analyzed. Finally, we construct a two-dimensional analogue [Formula: see text] by deleting, at each stage, the square whose coordinates simultaneously carry the respective forbidden digits. This set is an irregular version of the Sierpiński carpet. Its Hausdorff dimension is [Formula: see text] and its Lebesgue measure is zero. Several open problems and directions for further research are discussed.