[Paper Review] Excursions on Cantor-like Sets

Roberto DiMartino, Wilfredo Urbina|arXiv (Cornell University)|Nov 26, 2014

Mathematical Dynamics and Fractals4 references3 citations

TL;DR

This paper explores generalized Cantor-like sets beyond the classical ternary Cantor set, analyzing their topological, measure-theoretic, and fractal properties. It introduces constructions with variable removal proportions (e.g., $ C_ u $), demonstrates that such sets can be perfect, nowhere dense, uncountable, and have positive Lebesgue measure, and computes their Hausdorff dimension via scaling arguments, with applications to Riemann integrability counterexamples using fat Cantor sets.

ABSTRACT

The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article we will explore the richness, the peculiarities and the generalities that has $C$ and explore some variants and generalizations of it. For a more systematic treatment the Cantor like sets we refer to our previous paper.

Motivation & Objective

• To generalize the classical Cantor ternary set by varying the proportion of intervals removed at each stage.
• To analyze the topological properties (perfect, nowhere dense) and measure-theoretic behavior (positive measure) of these generalized sets.
• To compute the Hausdorff dimension of asymmetric and symmetric Cantor-like sets using scaling and self-similarity.
• To demonstrate the utility of fat Cantor sets in constructing counterexamples to the completeness of Riemann integrable functions.
• To provide a systematic foundation for Cantor-like sets, referencing prior work in [7] for deeper exploration.

Proposed method

• Constructs generalized Cantor sets $ C_ u $ by removing open intervals of length $ \lambda / 3^n $ from the center of each subinterval at step $ n $, preserving self-similarity.
• Uses ternary expansion characterization: $ C = \{ x \in [0,1] \mid \varepsilon_k(x) \in \{0,2\} \} $, generalizing to sets with restricted digit patterns.
• Applies the dimension trick: for a set with $ p $ self-similar parts each scaled by $ 1/m $, the Hausdorff dimension is $ d = \ln p / \ln m $.
• Derives the Lebesgue measure of $ C_\lambda $ as $ 1 - \lambda $, showing that $ C_\lambda $ has positive measure for $ 0 < \lambda < 1 $.
• Uses the Volterra set $ SVC(4) $ as a fat Cantor set with measure $ 1/2 $, and constructs a sequence of Riemann integrable functions converging to its indicator function.
• Employs the Lebesgue criterion: a function is Riemann integrable iff its set of discontinuities has measure zero, to show the limit function is not Riemann integrable.

Experimental results

Research questions

• RQ1 How do Cantor-like sets constructed by removing variable proportions of intervals differ in measure and dimension from the classical ternary Cantor set?
• RQ2 Can Cantor-like sets be both perfect and have positive Lebesgue measure, and if so, under what conditions?
• RQ3 What is the Hausdorff dimension of asymmetric Cantor-like sets, and how does it depend on the scaling and number of self-similar parts?
• RQ4 How can fat Cantor sets be used to construct a sequence of Riemann integrable functions whose pointwise limit is not Riemann integrable?
• RQ5 What is the role of ternary expansions in characterizing and generalizing Cantor-like sets?

Key findings

• The generalized Cantor set $ C_\lambda $ has Lebesgue measure $ 1 - \lambda $, so it has positive measure for $ 0 < \lambda < 1 $, making it a 'fat' Cantor set.
• The Hausdorff dimension of $ C_\lambda $ is $ d = \frac{\ln 2}{\ln 6 - \ln(3 - \lambda)} $, which depends on the removal parameter $ \lambda $.
• For symmetric Cantor-like sets with $ p $ self-similar parts each scaled by $ 1/m $, the Hausdorff dimension is $ d = \frac{\ln p}{\ln m} $, generalizing the classical case.
• The classical Cantor ternary set has Hausdorff dimension $ \frac{\ln 2}{\ln 3} \approx 0.6309 $, consistent with the general formula.
• The set of discontinuities of the limit function in the Riemann integrability counterexample is $ SVC(4) $, a fat Cantor set of measure $ 1/2 $, which has positive measure and thus violates Lebesgue's criterion.
• The sequence of indicator functions $ f_n $ of the removed intervals converges pointwise and in $ L^1 $ to the indicator of the fat Cantor set, but the limit is not Riemann integrable.