Properties of a random Cantor set with overlaps

2026/01/31 by Anna Chiara Lai, Paola Loreti
Mathematics · #math.NT

paper · pdf

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set Condition fails. Our methodology is based on the theory of expansions in non-integer bases.

Citations

• Counting base phi representations
• Intrinsic Diophantine Approximation for overlapping iterated function systems
• Proof of the exact overlaps conjecture for systems with algebraic contractions
• Overlapping iterated function systems from the perspective of Metric Number Theory
• Hausdorff Dimension of Planar Self-Affine Sets and Measures with Overlaps
• Sets of beta-expansions and the Hausdorff Measure of Slices through\n Fractals
• On the topology of polynomials with bounded integer coefficients
• Optimal expansions in non-integer bases
• Fractal Geometry: Mathematical Foundations and Applications.
• Representations for real numbers and their ergodic properties
• Additive functions of intervals and Hausdorff measure

Related

• A class of random Cantor sets
• Fractal uncertainty principle for random Cantor sets
• Random countable iterated function systems with overlaps and applications
• Random subsets of Cantor sets generated by trees of coin flips
• Geometry of measures in random systems with complete connections