Intersecting the twin dragon with rational lines (Q6648701)

The Knuth Twin Dragon is the set \N\[\N\left\{\sum_{k=1}^\infty\frac{d_k}{\alpha^k}:d_k\in\{0,1\}\right\},\N\]\Nwhere \(\alpha=-1+i\). The Hausdorff dimension of its boundary is \(s=(\log\lambda)/(\log\sqrt{2})\), where \(\lambda\) is the real numbers satisfying \(\lambda^3=\lambda^2+2\). In this paper, the intersections of the Knuth Twin Dragon with rational lines are investigated. By a result in [\textit{J. M. Marstrand}, Proc. Lond. Math. Soc. (3) 4, 257--302 (1954; Zbl 0056.05504)], the intersection with a generic line has Hausdorff dimension \(s-1\). This paper shows that the Hausdorff dimension of the intersection of the boundary of the Twin Dragon with rational lines is never equal to \(s-1\). Further, by revisiting results in [\textit{S. Akiyama} and \textit{K. Scheicher}, Acta Sci. Math. 71, No. 3--4, 555--580 (2005; Zbl 1111.11006)], uncountably many examples of horizontal, vertical, and diagonal lines are added.