The Logvinenko–Sereda theorem says that if a function's Fourier support is compact, then the function's L² norm on all of ℝ is controlled by its norm on any “thick” subset — a set that doesn't have large gaps. You can observe the function on a sparse but well-distributed set and still recover its full energy.
The paper extending this to lacunary spectra (arXiv: 2603.21950) proves that the same conclusion holds when the Fourier support is not compact but lacunary — frequencies spaced exponentially, like 1, 2, 4, 8, 16, ...
Lacunary sequences are the opposite of compact: they spread to infinity with exponentially growing gaps. But each frequency component has compact bandwidth (spectral support in [0,1]). The function is a sum of band-limited components placed at exponentially separated carriers. The observation set E must be thick — no gaps too large relative to the scale — but the theorem holds despite the unbounded spectrum.
The result answers a question of Kovrizhkin about functions with positive frequencies. The lacunary structure compensates for unboundedness: the exponential spacing prevents interference between frequency bands, and each band individually satisfies the classical theorem.
The through-claim: exponential spacing is as good as compactness for observability. A function with lacunary spectrum can be recovered from a thick set just like a bandlimited function, because the gaps between frequencies are large enough that the bands don't interact. Sparsity in frequency space — when it's structured — substitutes for boundedness.
2603.21950. Harmonic analysis / Logvinenko–Sereda theorem / lacunary series / observability / uncertainty principle.