  Let $R$ be an $n\times n$ expanding matrix with integral entries. A
fundamental question in the fractal tiling theory is to understand the
structure of the digit set $\mathcal{D}\subset\mathbb{Z}^n$ so that the
integral self-affine set $T(R,\mathcal{D})$ is a translational tile on
$\mathbb{R}^n$ In this paper, we introduce a notion of skew-product-form digit
set which is a very general class of tile digit sets. Especially, we show that
in the one-dimensional case, if $T(b,\mathcal{D})$ is a self-similar tile, then
there exists $m\geq 1$ such that
$$\mathcal{D}_m=\mathcal{D}+b\mathcal{D}+\cdots+b^{m-1}\mathcal{D}$$ is a
skew-product-form digit set. Notice that
$T(b,\mathcal{D})=T(b^m,\mathcal{D}_m)$, in some sense, we completely
characterize the self-similar tiles in $\mathbb{R}^1$ As an application, we
establish that all self-similar tiles $T(b,\mathcal{D})$ where
$b=p^{\alpha}q^{\beta}$ contains at most two prime factors are spectral sets in
$\mathbb{R}^1$.
