  For a partition of $[0,1]$ into intervals $I_1,\ldots,I_n$ we prove the
existence of a partition of $\mathbb{Z}$ into $\Lambda_1,\ldots, \Lambda_n$
such that the complex exponential functions with frequencies in $ \Lambda_k$
form a Riesz basis for $L^2(I_k)$, and furthermore, that for any
$J\subseteq\{1,\,2,\,\dots,\,n\}$, the exponential functions with frequencies
in $ \bigcup_{j\in J}\Lambda_j$ form a Riesz basis for $L^2(I)$ for any
interval $I$ with length $|I|=\sum_{j\in J}|I_j|$. The construction extends to
infinite partitions of $[0,1]$, but with size limitations on the subsets
$J\subseteq \mathbb{Z}$; it combines the ergodic properties of subsequences of
$\mathbb{Z}$ known as Beatty-Fraenkel sequences with a theorem of Avdonin on
exponential Riesz bases.
