Abstract
In connection to the Fuglede conjecture, we study groups of local translations associated to spectral sets, i.e., measurable sets in $\br$ or $\bz$ that have an orthogonal basis of exponential functions. We investigate the connections between the groups of local translations on $\bz$ and on $\br$ and present some examples for low cardinality. We present some relations between the group of local translations and tilings.
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