Positive-measure intersections of Bedford-McMullen carpets under similitudes

Abstract

Let $K=K(n,m,\Gamma )$ be a Bedford-McMullen carpet with $n>m\ge 2$ and $1<\#\Gamma <nm$, and let $\mu$ be a self-affine measure on $K$ with strictly positive weights. We prove that every similitude $f(z)=rOz+a$ satisfying $\mu (K\cap f(K))>0$ 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 $r\in n^{\mathbb{Q}}\cup m^{\mathbb{Q}}$ 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.

arXiv comment30 pages, 1 figure. v2: a new logarithmic commensurability theorem and some minor changes