Tiling and Spectrality for Generalized Sierpinski Self-Affine Sets

作者：Chen, Ming-Liang; Liu, Jing-Cheng^*; Zheng, Jia

来源：JOURNAL OF GEOMETRIC ANALYSIS, 2024, 34(1): 5.

DOI：10.1007/s12220-023-01447-y

• Sierpinski self-affine set
• Spectral set
• Translational tile
• Fuglede's conjecture

Let A is an element of M-2(Z) be an expanding integer matrix and D = {d(1) = 0, d(2), d(3)} subset of Z(2). It follows from Hutchinson (Indiana Univ Math J 30:713-747, 1981) that the generalized Sierpinski self-affine set T( A, D) is the unique compact set determined by the pair (A, D) satisfing the set-valued equation AT(A, D) = boolean OR(3)(i=1)(T(A, D) + d(i)). In this paper, we showthat Fuglede's conjecture holds on T(A, D), which states that T(A, D) is a spectral set if and only if T(A, D) is a translational tile. For the classical Sierpinski self-affine set T(A, D-c) with D-c = {(0, 0)(t), (1, 0)(t), (0, 1)(t)}, a finer characterization of tiling set is given. As an application, we find that the classical Sierpinski self-affine tile T(A, D-c) is suitable for Kolountzakis' conjecture on product domain. This enriches the results that are now known.