Jiang–Kong–Li–Wang and Simon–Taylor Cantor-dust questions

For $0<\lambda<\tfrac12$ , let $K_\lambda$ be the attractor of the iterated function system $\{x\mapsto\lambda x,\ x\mapsto\lambda x+1-\lambda\}$ , let $C_\lambda=K_\lambda\times K_\lambda$ , let $S^1=\{x\in\mathbb{R}^2:\|x\|=1\}$ , and define $E_\lambda=C_\lambda\cap S^1$ and $A_\lambda=C_\lambda+S^1$ . The circle-intersection problem asks for which parameters $\lambda$ the set $E_\lambda$ is infinite, and in particular whether $E_{1/3}$ is infinite. The Minkowski-sum problem asks for which parameters $\lambda$ the set $A_\lambda$ has nonempty interior in $\mathbb{R}^2$ , with the previously open range being $\tfrac14<\lambda<\tfrac13$ .

References

Primary source

arXiv

Additional references

Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle — arXiv

fractal-geometry harmonic-analysis

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed preprint claims to settle the Minkowski-sum question and substantially extend what is known about the circle intersection, but its conclusions have not been independently verified.

The entry concerns circle intersections and Minkowski sums of four-corner Cantor dusts, including the cases associated with $\lambda=1/3$ and the previously open parameter interval.

Known results

For $0<\lambda\le 2-\sqrt{3}$ , a preprint claims the circle intersection consists only of the two trivial points; it gives a negative answer at $\lambda=1/5$ .
The same work claims nontrivial intersections for $0.330384\le\lambda<1/2$ and continuum cardinality for $0.407493\le\lambda<1/2$ .
Du, Jiang, and Yao reportedly established, with computer assistance, at least $10{,}000{,}000$ intersection points at $\lambda=1/3$ .

September 2026 preprint

On September 8, 2026, a linked preprint claimed that the circle intersection is infinite in a new parameter range and that the Minkowski sum has interior exactly for $1/4<\lambda<1/2$ . This would address the cited questions, including $\lambda=1/3$ , but the preprint is unrefereed.

Current status (as of September 2026): The exact Minkowski-sum criterion and the new circle-intersection claims are asserted but unverified, so no verified complete resolution is recorded.

Sources

No solutions have been posted yet.