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Positive-measure intersections of Bedford-McMullen carpets under similitudes
Abstract
Let K=K(n,m,Γ) be a Bedford-McMullen carpet with n>m≥2 and 1<#Γ<nm, and let μ be a self-affine measure on K with strictly positive weights. We prove that every similitude f(z)=rOz+a satisfying μ(K∩ f(K))>0 has a diagonal or anti-diagonal orthogonal part O, without any arithmetic restrictions on the defining bases. Under the same intersection hypothesis, we also show that r∈ nℚ∪ mℚ whenever K is not a segment. For isometries satisfying this hypothesis, we further prove that f(K) is simply a translate of K, thereby establishing a strengthened form of the 2010 conjecture of Elekes, Keleti and Máthé in this planar self-affine context.
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BibTeX
@misc{hu2026positivemeasure,
title = {{Positive-measure intersections of Bedford-McMullen carpets under similitudes}},
author = {Ding-Kun Hu and Huo-Jun Ruan and Jian-Ci Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09277},
eprint = {2610.09277},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09277},
}
RIS
TY - UNPB
TI - Positive-measure intersections of Bedford-McMullen carpets under similitudes
AU - Ding-Kun Hu
AU - Huo-Jun Ruan
AU - Jian-Ci Xiao
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.09277
UR - https://arxiv.org/abs/2610.09277
AB - Let K=K(n,m,Γ) be a Bedford-McMullen carpet with n>m≥2 and 1<#Γ<nm, and let μ be a self-affine measure on K with strictly positive weights. We prove that every similitude f(z)=rOz+a satisfying μ(K∩ f(K))>0 has a diagonal or anti-diagonal orthogonal part O, without any arithmetic restrictions on the defining bases. Under the same intersection hypothesis, we also show that r∈ n^ℚ∪ m^ℚ whenever K is not a segment. For isometries satisfying this hypothesis, we further prove that f(K) is simply a translate of K, thereby establishing a strengthened form of the 2010 conjecture of Elekes, Keleti and Máthé in this planar self-affine context.
ER -