Discrete Problems in Harmonic Analysis and One Bit Sensing

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On the one hand, this project will advance one's understanding of currently important topics in signal processing, and on the other, it will stimulate interactions between two central areas of mathematics, analysis and number theory. In signal processing, so-called one-bit sensing replaces the usual linear measurements of a signal by incorporating the sign of the measurement. There are two motivations for this. First, the sign function is a basic type of nonlinearity, so results about it can represent a first step towards relaxing linearity in a number of significant settings. Second, in an important technological development, one-bit measurements can be done very frequently. Surprising experiments are using new methods that need theoretical justification, and in this context one-bit measurements have proven to be just as effective as linear ones. Topics in compressive sensing have an intimate relationship to questions in discrepancy theory, with one-bit sensing being the easiest place to uncover this relationship. This raises a new set of questions in discrepancy theory, and brings to compressive sensing new techniques. In particular, there are new tools to study lower bounds in the subject, which can reveal how effective the current random techniques are. Also, it can provide new approaches to sampling, such as the semi-random jittered sampling. In analysis and number theory, the project will study discrete variants of maximal oscillatory singular integrals. The latter, in the continuous case, are already fascinating objects. Passing to the discrete variants entails many additional complications that arise from subtle arithmetic structures. The results established in the project will be the first of this type, hence will require novel techniques. The new approach is a synthesis of breakthrough results of Bourgain and elemental methods in phase-plane analysis.