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$\mathbb {R}^{n}$
with consecutive digit sets
Let
$\{R_{k}\}_{k=1}^{\infty }$
be a sequence of expanding integer matrices in
$M_{n}(\mathbb {Z})$
, and let
$\{D_{k}\}_{k=1}^{\infty }$
be a sequence of finite digit sets with integer vectors in
${\mathbb Z}^{n}$
. In this paper, we prove that under certain conditions in terms of
$(R_{k},D_{k})$
for
$k\ge 1$
, the Moran measure
is a spectral measure. For the converse, we get a necessity condition for the admissible pair
$$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$
$(R,D)$
.
