Year: 2026
Type: preprint
Abstract: Let $K=K(n,m,Γ)$ be a Bedford-McMullen carpet with $n>m\geq2$ and $1<\#Γ0$ 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 $r\in n^{\mathbb{Q}}\cup m^{\mathbb{Q}}$ 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.
Source: arXiv (Cornell University)
Authors: Hu, Ding-Kun, Ruan, Huo-Jun, Xiao, Jian-Ci
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221062262