A function whose graph has positive doubling measure

TL;DR

This paper demonstrates that a doubling measure on the plane can assign positive measure to the graph of a continuous function, addressing a question about measure distribution on such graphs.

Contribution

It proves that doubling measures can give positive measure to continuous function graphs and that the doubling constant can be arbitrarily close to that of Lebesgue measure.

Findings

Doubling measures can assign positive measure to continuous function graphs.

The doubling constant can be made arbitrarily close to Lebesgue measure's constant.

Answers a previously open question by Wang, Wen, and Wen.

Abstract

We show that a doubling measure on the plane can give positive measure to the graph of a continuous function. This answers a question by Wang, Wen and Wen. Moreover we show that the doubling constant of the measure can be chosen to be arbitrarily close to the doubling constant of the Lebesgue measure.