2026/07/22 by Lai Jiang, Bing Li, Ruofan Li +1
#math.CA
Let E⊂ ℝ be a self-similar set generated by a homogeneous iterated function system Φ with contraction ratio ρ∈ (0,1). Assume that Φ satisfies the open set condition and dim\rm HE<1. Let f be a C1-diffeomorphism on ℝ. We prove that if log|f'(x)|/logρ\not∈ℚ for every x∈ E∩ f-1(E), then the upper Minkowski dimension of f(E)∩ E is strictly less than the Hausdorff dimension of E. We also establish a quantitative dimension drop result when E is a missing-digit set and f is an affine map with rational slope satisfying a certain arithmetic condition. Based on these results and a result of Shmerkin [Ann. of Math., 2019], we obtain characterizations of γ in various contexts such that dim\rm M((γE+α)∩ E)<dim\rm HE for every α∈ℝ.