  Generalizing a result given by Li, Miao and Wang in 2022, we study the
spectrality of a class of infinite convolutions in $\mathbb{R}^d$ generated by
sequences of nearly $d$-th power lattices. This allows us to easily construct
spectral measures with and without compact supports in $\mathbb{R}^d$.
According to a result on the relation between supports of infinite convolutions
and sets of infinite sums, we systematically study the Hausdorff and packing
dimensions of infinite sums of finite sets in $\mathbb{R}^d$. As an
application, we give concrete formulae for the Hausdorff and packing dimensions
of the supports of a class of spectral measures in $\mathbb{R}^d$ with the form
of infinite convolutions generated by specific sequences of nearly $d$-th power
lattices, and finally we deduce that there are spectral measures with and
without compact supports of arbitrary Hausdorff and packing dimensions in
$\mathbb{R}^d$.
