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

Abstract

The self-affine measure μM, D associated with an affine iterated function system {φ{symbol}d (x) = M- 1 (x + d)}d ∈ D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μM, D have been received much attention in recent years. One of the non-spectral problem on μM, D is to estimate the number of orthogonal exponentials in L2 (μM, D) and to find them. In the present paper we show that for an expanding integer matrix M ∈ M2 (Z) and the three-elements digit set D given byM = [(a, b; d, c)] and D = {((0; 0)), ((1; 0)), ((0; 1))}, if a c - b d ∉ 3 Z, then there exist at most 3 mutually orthogonal exponentials in L2 (μM, 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. © 2008 Elsevier Inc. All rights reserved.