Low autocorrelation sequences, difference sets, and flat polynomials

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The identification of binary sequences and arrays with small mean-squared aperiodic autocorrelation is an old problem originally motivated by its practical importance in communications engineering and condensed matter physics. Although the subject has its origin in an engineering problem, interesting questions in combinatorics, analysis, and number theory have emerged, some of which remain open since decades. There are also intimate connections to classical extremal polynomial questions, and as such the problem has been extensively studied by eminent mathematicians, such as Littlewood and Erdös. Recent progress has opened up new research directions, which will be explored within this project. The overall objective is to develop new methods for studying correlations of sequences and norms of polynomials and to use them to attack classical problems in the field and to unveil unifying principles. The general research methodology will be to combine methods from combinatorics and analytic number theory.