Work

Full measure universality for Cantor sets

Year: 2026

Type: article

Abstract: We investigate variants of the Erdős similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known fact that sets of positive Hausdorff dimension are not measure universal. We prove a weaker result for all Cantor sets A : there is a dense G δ set of full measure X ⊂ R d , such that for any bi-Lipschitz function f : R d → R d , the set of translations t such that f ( A ) + t ⊆ X is of measure zero. Equivalently, there is a null set B ⊂ R d such that R d ∖ ( f ( A ) + B ) is null for all bi-Lipschitz functions f .

Source: Advances in Mathematics

DOI: 10.1016/j.aim.2026.110978

Authors: Pablo Shmerkin, Alexia Yavicoli

Institution: University of British Columbia

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,790

Funder: Natural Sciences and Engineering Research Council of Canada