Graph-null sets

TL;DR

The paper investigates graph-null and translational Kakeya properties for plane sets, showing that among compact sets the translational Kakeya property, graph-null, and typically graph-null are equivalent. It proves that graphs of absolutely continuous functions are graph-null and that typical continuous functions have graph-null graphs, implying the existence of nowhere differentiable graphs with graph-null behavior, while also constructing a continuous function whose graph is not graph-null. The results distinguish graph-null from the translational Kakeya property in general, but align them for compact sets, and extend to absolutely continuous curves. The methods combine geometric-analytic approximations with measure-theoretic arguments to characterize when translations of a set touch zero area and to build explicit counterexamples.

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.

Theorems & Definitions (20)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.8
• Corollary 1.9
• Theorem 1.10
• Theorem 1.11