Given an analytic family F of curves in R^2, characterize when there exists a Borel set E⊂R^2 with Lebesgue measure |E|=0 such that, for every x∈[0,1]^2, there are a curve gamma∈F and a nontrivial arc I⊂gamma containing x for which I setminusx⊂ E.

Nikodym-type set and local smoothing conjectures

The problem asks when geometric families of curves admit Nikodym-type sets and how this relates to local-smoothing bounds for wave equations. The recent work is framed for analytic curve families in the plane and attributes the associated local-smoothing conjecture to Zahl.

Known results

Chang and Csörnyei: constructions show that strongly degenerate analytic families can admit Nikodym-type sets.
Wolff: an early local-smoothing result for sufficiently large $p$ .
Guth, Wang, and Zhang: established Sogge’s local-smoothing conjecture in two spatial dimensions for all $2<p<\infty$ .
A multi-parameter cinematic-curvature theorem gives $L^p$ maximal bounds above a dimension-dependent threshold and rules out the corresponding Nikodym obstruction.

September 2026 claimed classification

Alan Chang, Mingfeng Chen, Shaoming Guo, Minxing Shen, Tongou Yang, and Joshua Zahl report non-existence under a multi-parameter cinematic-curvature condition, examples in other settings, and a resolution of the cited local-smoothing problem. This is specialist preprint evidence of substantial progress, but the broad characterization remains unverified and is explicitly limited to the analytic-family framework.

Current status (as of October 2026): substantial progress is claimed for analytic planar curve families, while no verified resolution of the broader Nikodym-type set and local-smoothing problem is recorded.

Nikodym-type set and local smoothing conjectures

Primary source

Additional references

Progress summary

Solutions 0