On the dimension of $s$-Nikod\'ym sets

TL;DR

This paper establishes a bound on the Hausdorff dimension of certain Borel sets in the plane, showing that if each point is accessible via an s-dimensional family of lines, then the set's dimension is at most 2 minus s.

Contribution

It proves a new upper bound on the Hausdorff dimension of s-Nikodym sets in the plane based on their line accessibility properties.

Findings

Hausdorff dimension of N is at most 2 - s

Sets with line accessibility have constrained dimension

Provides a dimension bound for s-Nikodym sets

Abstract

Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.