A function whose graph has positive doubling measure (Q2789870)

A Borel regular measure \(\mu\) on \(\mathbb{R}^2\) is called doubling if for any adjacent (intersecting) squares \(Q_1\) and \(Q_2\) with the same side length, there exists a constant \(C < \infty\) such that \(0 < \mu(Q_1) \leq C\mu(Q_2)< \infty\). The authors show that for any \(\epsilon > 0\), there exist a doubling measure \(\mu\) on \([0, 1]^2\) with doubling constant less than \(1+\epsilon\) and a continuous function \(f : [0, 1] \to [0, 1]\) such that \(\mu(\mathrm{graph}(f)) > (1 - \epsilon)\mu([0, 1]^2)\). They construct such a measure using a 4-adic distribution of mass, changing the ``weight'' used in the distribution in the iteration process. Two figures illustrate their construction. Thus, the paper shows that a doubling measure on the plane can give positive measure to the graph of a continuous function. They argue that the graph \(E\) of any Lipschitz function is thin, i.e., \(\mu(E) = 0\) for every doubling measure \(\mu\) of \(\mathbb{R}^2\), thus answering a question by \textit{W. Wang} et al. [Ann. Acad. Sci. Fenn., Math. 38, No. 2, 535--546 (2013; Zbl 1294.28002)]. Using a result in [\textit{F. Peng} and \textit{S. Wen}, Nonlinearity 27, No. 6, 1287--1298 (2014; Zbl 1294.28001)], the authors also show that the doubling constant of the measure can be chosen arbitrarily close to the doubling constant of the Lebesgue measure.