Autocorrelation and flatness of height one polynomials

This thesis is concerned with two classes of polynomials whose height (meaning the largest absolute value of a coefficient) is 1: Littlewood polynomials, whose coefficients are +1 or - 1, and zero-one polynomials, whose coefficients are 0 or 1. We are interested in the behaviour of these polynomials on the unit circle in the complex plane. Roughly speaking, there is a tendency for a polynomial to be 'flat' on the unit circle if its autocorrelations are 'near zero', where the 'autocorrelations' can be regarded as dot products that measure the 'periodicity' of the coefficient sequence of the polynomial. In Chapter 1, we provide some illustrative conjectures as well as establishing some probabilistic language that is useful for studying the flatness or autocorrelations of 'typical' Littlewood polynomials or zero-one polynomials. In Chapter 2, we use properties of cosine sums to prove results about roots on the unit circle of Littlewood polynomials possessing certain kinds of symmetries. In particular, we prove that a type of Littlewood polynomial called a skewsymmetric Littlewood polynomial cannot have any roots on the unit circle. In Chapter 3, we show how one can compute all moments (meaning average values of powers) of autocorrelations of Littlewood polynomials, and we give an improved upper bound on the function that measures the minimum maximum autocorrelation (in absolute value) of a Littlewood polynomial. In Chapter 4, we give explicit formulae for the average fourth power of the fournorm of a zero-one polynomial, and show that this yields a surprising new proof of a known result about Sidon sets.