If X is a closed subset of the real line, denote by GX the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on X. In this paper we attempt to find GX in terms of X.
Alexei Poltoratski. "Spectral gaps for sets and measures." Acta Math. 208 (1) 151 - 209, 2012. https://doi.org/10.1007/s11511-012-0076-4