Graph-null sets
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915767303274496 |
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| author | Laczkovich, M. Máthé, A. |
| author_facet | Laczkovich, M. Máthé, A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_01195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graph-null sets Laczkovich, M. Máthé, A. Combinatorics Classical Analysis and ODEs 28A75 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. |
| title | Graph-null sets |
| topic | Combinatorics Classical Analysis and ODEs 28A75 |
| url | https://arxiv.org/abs/2602.01195 |