  It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there
exist spectral measures with arbitrary Hausdorff dimensions, and it is natural
to pose the question of whether similar phenomena occur for other dimensions of
spectral measures. In this paper, we first obtain the formulae of Assouad
dimension and of lower dimension for a class of Moran measures in dimension one
that is introduced by An and He [J. Funct. Anal. 266 (2014), 343--354]. Based
on these results, we show the existence of spectral measures with arbitrary
Assound dimensions $\dim_A$ and lower dimensions $\dim_L$ ranging from $0$ to
$1$, including non-atomic zero-dimensional spectral measures and
one-dimensional singular spectral measures, and prove that the two values may
coincide. In fact, more is obtained that for any $0 \leq t \leq s \leq r \leq
u\leq 1$, there exists a spectral measure $\mu$ such that \[\dim_L \mu=t,
\dim_H \mu=s, \dim_P\mu=r~ \text{and} \dim_A\mu=u,\] where $\dim_H$ and
$\dim_P$ denote the Hausdorff dimension and packing dimension of the measure
$\mu$, respectively. This result improves and generalizes the result of Dai and
Sun more simply and flexibly.
