Non-Spectral Problem of Self-Affine Measures in R³

Abstract

The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on μ M,D is to estimate the number of orthogonal exponentials in L 2 ( μ M,D ) . In the present paper, we establish some relations inside the zero set by the Fourier transform of the self-affine measure μ M,D . Based on these facts, we show that μ M,D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L 2 ( μ M,D) , where the number 4 is the best possible. This extends several known conclusions.