TY - CHAP
T1 - Some Variants of Orponen’s Theorem on Visible Parts of Fractal Sets
AU - Matheus, Carlos
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - It was recently established by T. Orponen that the visible parts from almost every direction of a compact subset of ℝn have Hausdorff dimension at most n−150n. In this note, we refine Orponen’s argument in order to show that the visible parts from almost every direction of a compact subset of ℝn have Hausdorff dimension at most n−min{15,1n+2}. Moreover, we also show that some classes of dynamically defined Cantor sets K⊂ ℝn with Hausdorff dimension d>max{3,(n−1)+(n−1)(n+3)2} have visible parts of Hausdorff dimension at most max{3d+3d+3,(n+1)d+(n−1)d+2} from almost every direction.
AB - It was recently established by T. Orponen that the visible parts from almost every direction of a compact subset of ℝn have Hausdorff dimension at most n−150n. In this note, we refine Orponen’s argument in order to show that the visible parts from almost every direction of a compact subset of ℝn have Hausdorff dimension at most n−min{15,1n+2}. Moreover, we also show that some classes of dynamically defined Cantor sets K⊂ ℝn with Hausdorff dimension d>max{3,(n−1)+(n−1)(n+3)2} have visible parts of Hausdorff dimension at most max{3d+3d+3,(n+1)d+(n−1)d+2} from almost every direction.
UR - https://www.scopus.com/pages/publications/85116904245
U2 - 10.1007/978-3-030-74863-0_16
DO - 10.1007/978-3-030-74863-0_16
M3 - Chapter
AN - SCOPUS:85116904245
T3 - Lecture Notes in Mathematics
SP - 517
EP - 533
BT - Lecture Notes in Mathematics
PB - Springer Science and Business Media Deutschland GmbH
ER -