Level sets of the Takagi function: Hausdorff dimension
The Takagi function \(\tau\) is a continuous, nowhere differentiable function from the unit interval \([0,1]\) to the reals \(\mathbb R\). The properties of its graph \[ {\mathcal G}_\tau:=\{(x,y)\in\mathbb R^2:\tau(x)=y\} \] and its level sets have been the object of some investigations in the recent past: for instance, \textit{P. C. Allaart} and \textit{K. Kawamura} [Math. Proc. Camb. Philos. Soc. 140, No. 2, 269--295 (2006; Zbl 1086.26004)] have proved that the Hausdorff dimension of \({\mathcal G}_\tau\) is 1; \textit{J. B. Brown} and \textit{G. Kozlowski} [Am. Math. Mon. 110, No. 2, 142--147 (2003; Zbl 1050.26004)] showed that \({\mathcal G}_\tau\) is lipschitzian of any order strictly less than 1; \textit{Y. Baba} [Proc. Am. Math. Soc. 91, 373--376 (1984; Zbl 0518.26004)] proved that the level set at height \(2/3\) has Hausdorff dimension \(1/2\). Here, the author extends Baba's result to intersections of \({\mathcal G}_\tau\) with lines in the plane of arbitrary integer slope. Precisely, exploiting the self-similarity of \({\mathcal G}_\tau\) he shows in Theorem 1.1 that for any integer \(k\) there exists a line of slope \(k\) whose intersection with \({\mathcal G}_\tau\) has both Hausdorff and Minkowski dimensions equal to \(1/2\). Further, in Theorem 1.2 the author shows that the Minkowski dimension of the intersection of \({\mathcal G}_\tau\) with any line of integer slope is less than \(0.668\) (such a constant has been determined by means of a combinatorial argument together with computer calculations). His conjecture is that Theorem 1.2 can be improved by replacing this upper bound with the best possible constant, namely \(1/2\). The author leaves as an open problem to estimate the Hausdorff and Minkowski dimensions of \({\mathcal G}_\tau \cap L\), \(L\) being an arbitrary line in the plane.
scientific article; zbMATH DE number 5708703
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