Extend restricted projection theorems beyond smooth curves

Extend restricted projection theorems from length measures on smooth curves in Grassmannians to other, potentially singular, measures on the Grassmannian.

Background

The paper discusses restricted Marstrand-type projection theorems in which the family of projection planes is sampled from a singular measure rather than from the natural Lebesgue measure on the Grassmannian. Existing results for smooth curves rely substantially on smoothness, while the paper develops a spread-outness framework that applies to broader classes of measures.

The authors explicitly identify the extension of smooth-curve arguments to other measures as unresolved. This problem concerns determining sufficient geometric or measure-theoretic hypotheses under which almost every projection with respect to a singular Grassmannian measure has the expected Hausdorff dimension.

References

These proofs rely on smoothness, and it is not clear how to extend them to other measures.

— Spread-out Furstenberg sets II: general configurations and fractal restricted projections  (2610.13682 - Cohen et al., 9 Oct 2026) in Section 1, subsection “Restricted projections”

All the proofs in this line of work rely on the smoothness of the curve, and it is not clear how to extend them to more general measures.

— Spread-out Furstenberg sets I: the sticky case  (2610.13677 - Orponen et al., 9 Oct 2026) in Section 1, subsection “Restricted projections”