We consider some parameterized planar sets with unbounded digits. We investigate these sets by using the method of “transversality", which is the main tool in investigating self-similar sets with overlaps. We calculate the Hausdorff dimension of these sets for typical parameters in some region with respect to the 2-dimensional Lebesgue measure. In addition, we estimate the local dimension of the exceptional set of parameters.
The author would like to express his gratitude to Professor Hiroki Sumi and Kanji Inui for their valuable comments. The author also would like to express his gratitude to the reviewer for valuable comments. This study is strongly motivated by the Ph.D. Thesis of Kanji Inui. Figure 1 is made by “Fractal Gazer," which was developed by Masaaki Wada. This study is supported by JSPS KAKENHI Grant Number JP 19J21038.
"The Hausdorff dimension of some planar sets with unbounded digits." Osaka J. Math. 59 (4) 755 - 776, October 2022.