The self-affine measure μ M,D associated with an affine iterated function system {φ d (x) = M −1 (x + d)} d∈D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μ M,D have been received much attention in recent years. One of the non-spectral problem on μ M,D is to estimate the number of orthogonal exponentials in L 2 (μ M,D ) and to find them. In the present paper we show that for an expanding integer matrix M ∈ M 2 (Z) and the three-elements digit set D given byif ac − bd / ∈ 3Z, then there exist at most 3 mutually orthogonal exponentials in L 2 (μ M,D ), and the number 3 is the best. This confirms the three-elements digit set conjecture on the non-spectrality of self-affine measures in the plane

The self-affine measure μ M,D corresponding to an expanding matrix M ∈ M n (R) and a finite subset D ⊂ R n is supported on the attractor (or invariant set) of the iterated function system {φ d (x) = M −1 (x + d)} d∈D . The spectral and non-spectral problems on μ M,D , including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μ M,D is to estimate the number of orthogonal exponentials in L 2 (μ M,D ) and to find them. In the present paper we show that if a, b, c ∈ Z, |a| > 1, |c| > 1 and ac ∈ Z \ (3Z),then there exist at most 3 mutually orthogonal exponentials in L 2 (μ M,D ), and the number 3 is the best. This extends several known conclusions. The proof of such result depends on the characterization of the zero set of the Fourier transformμ M,D , and provides a way of dealing with the non-spectral problem.

In this paper, we consider the non-spectral problem for the planar self-affine measures µ M,D generated by an expanding integer matrix M ∈ M 2 (Z) and a finite digitand gcd(det(M), p) = 1, then there exist at most p 2 mutually orthogonal exponential functions in L 2 (µ M,D ). In particular, if p is a prime, then the number p 2 is the best.

Let R be an n × n expanding matrix with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set D ⊂ Z n so that the integral self-affine set T (R, D) is a translational tile on R n . In this paper, we introduce a notion of skewproduct-form digit set which is a very general class of tile digit sets. Especially, we show that in the one-dimensional case, if T (b, D) is a self-similar tile, then there exists m ≥ 1 such that, in some sense, we completely characterize the self-similar tiles in R 1 . As an application, we establish that all self-similar tiles T (b, D) where b = p α q β contains at most two prime factors are spectral sets in R 1 .

No abstract