Abstract: In this talk the following problem is considered: How much of an integral norm of a function with compactly supported Fourier transform can be concentrated on a sparse subset of the real line? The inequalities that form an answer to this question turn out to be the main building blocks for applications from signal processing and analytic number theory. I will describe some of the applications and will outline existing strategies (building on work of Selberg and of Donoho and Logan) to prove these inequalities.