Non-spectral problem for the planar self-affine measures

TL;DR

This paper investigates the spectral properties of planar self-affine measures generated by integer matrices and digit sets, establishing bounds on orthogonal exponential functions under specific algebraic and geometric conditions.

Contribution

It provides new conditions under which the number of mutually orthogonal exponential functions in the measure's L^2 space is bounded, especially when the determinant and a prime number are coprime.

Findings

Maximum of p^2 orthogonal exponentials when gcd(det(M), p)=1

p^2 is the optimal bound for prime p

Conditions linking the zero set of Fourier transform to orthogonality

Abstract

In this paper, we consider the non-spectral problem for the planar self-affine measures $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^2$. Let $p\geq2$ be a positive integer, $E_p^2:=\frac{1}{p}\{(i,j)^t:0\leq i,j\leq p-1\}$ and $\mathcal{Z}_{D}^2:=\{x\in[0, 1)^2:\sum_{d\in D}{e^{2\pi i\langle d,x\rangle}}=0\}$. We show that if $\emptyset\neq\mathcal{Z}_{D}^2\subset E_p^2\setminus\{0\}$ and $\gcd(\det(M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2(\mu_{M,D})$. In particular, if $p$ is a prime, then the number $p^2$ is the best.