AI 中文总结
本文研究平面Nikodym型集合的存在性，通过新极大函数和局部平滑估计以及Falconer投影定理的非线性推广，结合驯顺几何，精确刻画了解析曲线族中Nikodym型集合的存在条件。
AI 中文摘要
Nikodym在平面中构造了一个测度为零的Borel集合，该集合包含穿过单位正方形中每个点的带孔线段。Fenton和Falconer提出了一个问题：当“过点的直线”被替换为“具有给定圆心的圆或圆弧”时，是否可能进行类似的构造？Bourgain和Marstrand对此给出了否定回答。我们研究了这个问题的推广，其中“过点的直线”或“具有给定圆心的圆”被替换为更一般的平面曲线解析族。作为关键成分，我们证明了满足Sogge电影曲率条件的多参数类似物的曲线族的新极大函数和局部平滑估计；这使我们能够在某些特定情况下建立Nikodym型集合的不存在性。该结果还解决了第六作者提出的一个局部平滑猜想。在另一个方向上，我们利用Falconer给定投影定理的一个最近非线性推广，在其他特定情况下构造了Nikodym型集合。将这些成分与驯顺几何的工具相结合，我们精确刻画了对于平面曲线的解析族，Nikodym型集合何时存在。
英文摘要
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.
