  In a former paper we introduced the concept of localisation of ideals in the
Fourier algebra of a locally compact Abelian group. It turns out that
localisability of a closed ideal in the Fourier algebra is equivalent to the
synthesisability of the annihilator of that closed ideal which corresponds to
this ideal in the measure algebra. This equivalence provides an effective tool
to prove synthesisability of varieties on locally compact Abelian groups. In
another paper we used this method to show that when investigating
synthesisability of a variety, roughly speaking, compact elements of the group
can be neglected. Using these results, in this paper we completely characterise
those locally compact Abelian groups on which spectral synthesis holds.
