Spectrality of Self-Similar Tiles

Abstract

We call a set $$K \subset {\mathbb {R}}^s$$ with positive Lebesgue measure a spectral set if $$L^2(K)$$ admits an exponential orthonormal basis. It was conjectured that K is a spectral set if and only if K is a tile (Fuglede’s conjecture). Although the conjecture was proved to be false in $${\mathbb {R}}^s$$ , $$s\ge 3$$ (Kolountzakis in Forum Math 18:519–528, 2006; Tao in Math Res Lett 11:251–258, 2004), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties of a class of self-similar tiles K in $${\mathbb {R}}$$ that has a product structure on the associated digit sets. We show that strict product-form tiles and the associated modulo product-form tiles are spectral sets. For the converse question, we give a pilot study for the self-similar set K generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality and verify Fuglede’s conjecture for this special case. We also make use of this case to illustrate the theorems and discuss some questions that arise in the general situation.