Abstract

Let
${\mu}_{M,D}$
be the planar selfaffine measure generated by an expansive integer matrix
$M\in {M}_{2}(\mathbb{Z})$ and a
noncollinear integer digit set
$$D=\left\{\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)\phantom{\rule{0.17em}{0ex}},\left(\begin{array}{c}\hfill {\alpha}_{1}\hfill \\ \hfill {\alpha}_{2}\hfill \end{array}\right)\phantom{\rule{0.17em}{0ex}},\left(\begin{array}{c}\hfill {\beta}_{1}\hfill \\ \hfill {\beta}_{2}\hfill \end{array}\right)\phantom{\rule{0.17em}{0ex}},\left(\begin{array}{c}\hfill {\alpha}_{1}{\beta}_{1}\hfill \\ \hfill {\alpha}_{2}{\beta}_{2}\hfill \end{array}\right)\right\}.$$ 
We show that
${\mu}_{M,D}$
is a spectral measure if and only if there exists a matrix
$Q\in {M}_{2}(\mathbb{R})$ such that
$(\stackrel{~}{M},\stackrel{~}{D})$ is admissible,
where
$\stackrel{~}{M}=QM{Q}^{1}$ and
$\stackrel{~}{D}=QD$. In particular,
when
${\alpha}_{1}{\beta}_{2}{\alpha}_{2}{\beta}_{1}\notin 2\mathbb{Z}$,
${\mu}_{M,D}$ is a spectral measure
if and only if
$M\in {M}_{2}(2\mathbb{Z})$.
This completely settles the spectrality of the selfaffine measure
${\mu}_{M,D}$.

Keywords
selfaffine measure, spectral measure, spectrum, admissible

Mathematical Subject Classification
Primary: 28A25, 28A80
Secondary: 42C05, 46C05

Milestones
Received: 17 September 2023
Revised: 8 December 2023
Accepted: 22 January 2024
Published: 18 February 2024

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