  Let $F$, $S$ be bounded measurable sets in $\mathbb{R}^d$. Let $P_F :
L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d) $ be the orthogonal projection
on the subspace of functions with compact support on $F$, and let $B_S :
L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ be the orthogonal projection
on the subspace of functions with Fourier transforms having compact support on
$S$. In this paper, we derive improved distributional estimates on the
eigenvalue sequence $1 \geq \lambda_1(F,S) \geq \lambda_2(F,S) \geq \cdots > 0$
of the \emph{spatio-spectral limiting operator} $B_S P_F B_S :
L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$. The significance of such
estimates lies in their diverse applications in medical imaging, signal
processing, geophysics and astronomy.
  Our proof is based on the decomposition techniques developed in
\cite{MaRoSp23}. The novelty of our approach is in the use of a two-stage
dyadic decomposition with respect to both the spatial and frequency domains,
and the application of the results in \cite{ArieAzita23} on the eigenvalues of
spatio-spectral limiting operators associated to cubical domains.
