By discrete here, I just mean that it's a linear combination of (potentially infinitely many) delta diracs, with discrete support.
Say a discrete subset of $\mathbb{R}$ is semiperiodic if it's dirac measure satisfies the condition in the title; i.e. it's Fourier transform is a linear combination of delta diracs with discrete support in $\mathbb{C}$.
It's clear that every periodic set is semiperiodic (their dirac measure is a dirac comb), and the semiperiodic sets are closed under finite union. From this, one might conject that every semiperiodic set is a countable union of periodic sets, but strangely enough, an argument involving RH can show that this is unlikely; in particular, assuming RH and that the nontrivial zeroes of RZF are linearly independent over $\mathbb{Q}$, which are both thought likely (but unproven), they form a semiperiodic set which is not the countable union of periodic sets. There are probably unconditional examples of this as well but unfortunately I have difficulties providing constructions of semiperiodic sets apart from using dirac combs.
The structure of these sets is quite fragile; changing semiperiodic sets in small ways causes them to be no longer semiperiodic. If you add a single point to a semiperiodic set, it is no longer semiperiodic as that single point has a continuous Fourier transform which is nonzero everywhere.
Additionally, since the Fourier transform seems to be concerned with "periodic behaviours", it'd be hard to believe that you could nudge a single point in a semiperiodic set in either direction without the resulting set ceasing to be semiperiodic.
Is there some geometric insight to this that I'm missing? I'm struggling to find a pattern in these sets; they seem so rigid and immutable yet so elusive as well. What other methods are there to construct these sets, and is there some nice way to parametrize them?