We present a sufficient condition on sets E and F in \({\mathbb R}^d\) to ensure compactness of Fourier concentration operators by introducing the notion of sets which are very thin at infinity. We are able to show that if the sets E and F are both very thin at infinity, then the associated Fourier concentration operator is compact on \(L^2({\mathbb R}^d)\) . The proof relies on a combination of the Logvinenko-Sereda uncertainty principle together with an uncertainty principle due to Shubin, Vakilian and Wolff. This provides a partial answer to a question posed by Katsnelson and Machluf on truncated Fourier operators.
