article · arXiv (Cornell University) · 2026

Graph-null sets

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Abstract

We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $λ_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy $A'$ of $A$ and for every $ε>0$, there is a finite sequence of vertical and horizontal translations bringing $A$ to $A'$ such that the area touched during the horizontal translations is less than $ε$. These properties are equivalent if $A$ is compact. We show that the graph of every absolutely continuous function is graph-null. Also, the graph of a typical continuous function is graph-null. Therefore, there are nowhere differentiable continuous functions whose graphs are graph-null. Still, we show that there exists a continuous function whose graph is not graph-null.

Cite this paper

APA 7

Laczkovich, M., & Máthé, A. (2026). Graph-null sets. arXiv (Cornell University).

MLA 9

Laczkovich, M., and A. Máthé. “Graph-null sets.” arXiv (Cornell University), 2026.

Chicago

Laczkovich, M., and A. Máthé. 2026. “Graph-null sets.” arXiv (Cornell University).

Harvard

Laczkovich, M. and Máthé, A. (2026) ‘Graph-null sets’, arXiv (Cornell University).

BibTeX

@article{laczkovich2026graph,
  title = {{Graph-null sets}},
  author = {Laczkovich, M. and Máthé, A.},
  journal = {arXiv (Cornell University)},
  year = {2026}
}

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