Graph-null sets

TL;DR

This paper introduces the concept of graph-null sets in the plane, explores their properties, and demonstrates that many graphs of continuous functions are graph-null, including some nowhere differentiable functions, but not all.

Contribution

It defines the notion of graph-null sets, establishes their equivalence with the translational Kakeya property for compact sets, and characterizes graphs of various classes of functions as graph-null.

Findings

Graphs of absolutely continuous functions are graph-null.

Typical continuous functions have graph-null graphs.

There exist continuous functions whose graphs are not graph-null.

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 $\lambda_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy $A'$ of $A$ and for every $\epsilon >0$, 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 $\epsilon$. 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.