Yu's nonempty-interior conjectures for radial projections and division sets

Yu's conjectures assert, for the relevant pairs of Cantor sets and their radial projections or division sets, that the corresponding sets have nonempty interior. The supplied sources do not state the precise hypotheses or definitions of the two conjectures sufficiently to give a more exact quantified formulation. In the initial-block family $K_{a,m}$ and $K_{b,n}$ , with $c_a=m/(a-1)$ and $c_b=n/(b-1)$ , the reported result claims that every radial projection of $K_{a,m}\times K_{b,n}$ has nonempty interior when $c_a+c_b\geq 1$ , while for multiplicatively independent bases and $c_a+c_b<1$ some radial images, including the relevant cross-division set, are compact and nowhere dense.

References

Primary source

arXiv

Additional references

A thickness boundary and modular obstructions for two-set radial projections — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new paper claims to settle the conjectures for a broad explicit family, proving a sharp threshold and giving counterexamples below it.

The problem concerns Yu's conjectures that certain radial projections and division sets have nonempty interior. The reported result treats a specified initial-block, multiplicatively independent family, not arbitrary Cantor sets or observers.

September 2026 threshold and counterexamples

The paper claims that every observer gives nonempty interior when $c_a+c_b\ge 1$ . Under multiplicative independence and $c_a+c_b<1$ , it constructs open observer sets whose radial images are compact and nowhere dense, thereby claiming counterexamples to two conjectured interior conclusions. The claim is unverified.

Current status (as of September 2026): The conjectures are claimed resolved for the specified family, with a threshold at $c_a+c_b=1$ and counterexamples below it; verification remains outstanding, and the general case remains open.

Sources

No solutions have been posted yet.