Arbitrarily sparse spectra for self-affine spectral measures

TL;DR

This paper demonstrates that self-affine spectral measures generated by certain matrices and digit sets can have arbitrarily sparse spectra with zero Beurling dimension, expanding understanding of spectral properties in fractal measures.

Contribution

It proves that for digit sets smaller than the determinant, the associated spectral measures can possess arbitrarily sparse spectra with zero Beurling dimension.

Findings

Spectral measures exist when B<|det(R)|.

Spectra can be constructed to be arbitrarily sparse.

Spectral measures can have zero Beurling dimension.

Abstract

Given an expansive matrix $R\in M_d({\mathbb Z})$ and a finite set of digit $B$ taken from $ {\mathbb Z}^d/R({\mathbb Z}^d)$. It was shown previously that if we can find an $L$ such that $(R,B,L)$ forms a Hadamard triple, then the associated fractal self-affine measure generated by $(R,B)$ admits an exponential orthonormal basis of certain frequency set $\Lambda$, and hence it is termed as a spectral measure. In this paper, we show that if #$B<|\det (R)|$, not only it is spectral, we can also construct arbitrarily sparse spectrum $\Lambda$ in the sense that its Beurling dimension is zero.