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Elekes–Keleti–Máthé conjecture on intersections of Bedford–McMullen carpets

Let K=K(n,m,Γ)⊂R2K=K(n,m,\Gamma)\subset\mathbb{R}^{2} be a Bedford–McMullen carpet with n>m≥2n>m\ge 2 and 1<#Γ<nm1<\#\Gamma<nm. For every isometry f:R2→R2f:\mathbb{R}^{2}\to\mathbb{R}^{2}, if dim⁡H(K∩f(K))=dim⁡HK\dim_{\mathrm{H}}\bigl(K\cap f(K)\bigr)=\dim_{\mathrm{H}}K, then f(K)f(K) and KK differ only by a translation.

References

Primary source

arXiv

Additional references

dynamical-systemsfractal-geometrygeometric-measure-theory
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Progress summary

Refreshed
Claimed progress

A new preprint reports stronger rigidity for positive-measure intersections of these carpets, but the full conjecture remains open.

The Elekes–Keleti–Máthé conjecture concerns rigidity of intersections between Bedford–McMullen carpets under geometric maps. The retrieved literature contains related slicing, embedding, and intersection-dimension results, but no proof or disproof of the conjecture in its full stated form.

October 2026 positive-measure intersection result

Ding-Kun Hu, Huo-Jun Ruan, and Jian-Ci Xiao report that arithmetic restrictions on the bases can be removed for their similitude theorem on positive-measure intersections of Bedford–McMullen carpets, with a stronger translation conclusion for isometries. This is substantive progress toward the conjectured rigidity, but it is limited to the Bedford–McMullen setting and positive-measure intersections, and the claim is unverified.

Current status (as of October 2026): A claimed unverified advance establishes stronger rigidity for positive-measure Bedford–McMullen carpet intersections, while the full Elekes–Keleti–Máthé conjecture remains open.

Sources

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