Non-spectrality of planar self-affine measures with three-elements digit set

Abstract

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

Spectrality of the planar Sierpinski family

“…The following result is known among [7,[20][21][22][23][24][25]. Here we give a simple proof of it for convenience of readers.…”

The cardinality of certain μM,D-orthogonal exponentials

“…More recently, the author [10][11][12] proved Conjecture 1 for a class of planar self-affine measures with three-elements digit set. Conjecture 1 is still open for the four-elements digit set, even in the following cases:…”

Non-spectral self-affine measure problem on the plane domain

“…Based on these established facts, we prove Theorem 1.3 in Section 4. It is worth noting this is different from the method of [22,34]. But it is difficult to find any general principles for dealing with similar non-spectral questions.…”

“…J.-L. Li [21] proved that any set of μ M,D -orthogonal exponentials contains at most 3 elements, and the number 3 is the best. More recently, J.-L. Li [22] proved that for the self-affine measure μ M,D corresponding to…”

“…(See [22].) For an expanding integer matrix M ∈ M n (Z) and a finite digit set D ⊆ Z n , if |D| / ∈ W (m), then μ M,D is a non-spectral measure and the non-spectral problem on this μ M,D falls in the class (I).…”