Polynomial Norm Problems in Number Theory

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In this proposal, we study polynomials with restricted coefficients. In particular, we are interested in  the following classes of polynomials: Unimodular Polynomials, Littlewood Polynomials and Newman Polynomials, which are the polynomials whose coefficients are respectively complex numbers on unit circle, from {-1,+1} and from {0, +1}. In this project, we are interested in studying the distributions of Lp norms over the unit circle among these classes of polynomials for various non-negative p.*******In the study of polynomials, one of the most important results is Kahane's theorem on the ultra-flat unimodular polynomials. Kahane's theorem states that there are sequences of unimodular polynomials {Pn} and of real numbers {En} such that (1): 1-En 1/20 such that for any sequence of Littlewood polynomials {Pn} we have (1+c)n1/2 < L4(Pn). The L4 Norm Conjecture clearly implies Erdos' conjecture. Moreover, our conjecture also implies the non-existence of Barker sequences with even length. This conjecture suggests finding Littlewood polynomials with small Lp norm.*******Very recently, Erdelyi and the proposer showed that the limiting distribution for the L4 norm of Littlewood polynomials is the one-point distribution. This result gives us plenty of Littlewood polynomials with the L4 norm close to its mean value. We plan to extend this result to other subclasses of Littlewood polynomials and hope to find smaller mean values of the L4 norm.  This will shed light on the L4 norm conjecture.*** ***Mahler measure of a polynomial <