  Let $G$ be a closed subgroup of ${\mathbb R}^d$ and let $\nu$ be a Borel
probability measure admitting a Riesz basis of exponentials with frequency sets
in the dual group $G^{\perp}$. We form a multi-tiling measure $\mu =
\mu_1+...+\mu_N$ where $\mu_i$ is translationally equivalent to $\nu$ and
different $\mu_i$ and $\mu_j$ have essentially disjoint support. We obtain some
necessary and sufficient conditions for $\mu$ to admit a Riesz basis of
exponentials . As an application, the square boundary, after a rotation, is a
union of two fundamental domains of $G = {\mathbb Z}\times {\mathbb R}$ and can
be regarded as a multi-tiling measure. We show that, unfortunately, the square
boundary does not admit a Riesz basis of exponentials of the form as a union of
translate of discrete subgroups ${\mathbb Z}\times \{0\}$. This rules out a
natural candidate of potential Riesz basis for the square boundary.
