On planar Nikodym-type sets
Abstract
Nikodym constructed a measure-zero Borel set in the plane that contains a punctured line segment passing through each point in the unit square. Fenton and Falconer asked whether a similar construction is possible when ``lines through a point'' are replaced by ``circles or circular arcs with a prescribed center.'' This was answered in the negative by Bourgain and Marstrand. We study generalizations of this question, in which ``lines through a point'' or ``circles with prescribed center'' are replaced by more general analytic families of plane curves. As a key ingredient, we prove new maximal function and local smoothing estimates for families of curves satisfying a multi-parameter analogue of Sogge's cinematic curvature condition; this allows us to establish the non-existence of Nikodym-type sets in certain specific settings. The result also resolves a local smoothing conjecture by the sixth author. In the other direction, we use a recent nonlinear generalization of Falconer's prescribed projection theorem to construct Nikodym-type sets in other specific settings. Combining these ingredients with tools from tame geometry, we precisely characterize when Nikodym-type sets exist for analytic families of plane curves.
Related
- Pinned distances of planar sets with low dimension
- On the minimizer of the barycentric isoperimetric ratio among connected planar sets
- Almost Linear Universal Point Sets for Planar Graphs
- Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results
- Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
BibTeX
@misc{chang2026planar,
title = {{On planar Nikodym-type sets}},
author = {Alan Chang and Mingfeng Chen and Shaoming Guo and Minxing Shen and Tongou Yang and Joshua Zahl},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39217},
eprint = {2609.39217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39217},
}
RIS
TY - UNPB
TI - On planar Nikodym-type sets
AU - Alan Chang
AU - Mingfeng Chen
AU - Shaoming Guo
AU - Minxing Shen
AU - Tongou Yang
AU - Joshua Zahl
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39217
UR - https://arxiv.org/abs/2609.39217
AB - Nikodym constructed a measure-zero Borel set in the plane that contains a punctured line segment passing through each point in the unit square. Fenton and Falconer asked whether a similar construction is possible when ``lines through a point'' are replaced by ``circles or circular arcs with a prescribed center.'' This was answered in the negative by Bourgain and Marstrand. We study generalizations of this question, in which ``lines through a point'' or ``circles with prescribed center'' are replaced by more general analytic families of plane curves. As a key ingredient, we prove new maximal function and local smoothing estimates for families of curves satisfying a multi-parameter analogue of Sogge's cinematic curvature condition; this allows us to establish the non-existence of Nikodym-type sets in certain specific settings. The result also resolves a local smoothing conjecture by the sixth author. In the other direction, we use a recent nonlinear generalization of Falconer's prescribed projection theorem to construct Nikodym-type sets in other specific settings. Combining these ingredients with tools from tame geometry, we precisely characterize when Nikodym-type sets exist for analytic families of plane curves.
ER -