Energy spreading polynomials

As is well known in the engineering world, a polynomial is a very useful model for the response (in receive mode) or output (in transmit mode) of a line array of equally spaced antenna elements. In some applications one wishes to transmit or receive omnidirectionally while simultaneously using all antenna elements at their full power. This is equivalent to a classical question in mathematical analysis, first formally posed by Littlewood in 1966 but considered by Hardy and Littlewood in 1913. Namely, if a polynomial has coefficients of equal modulus, how close to constant can its modulus be on the unit circle? We examine various attempts to answer this question since Littlewood first presented it, and explore possible interpretations and applications of such “energy spreading polynomials” in the mathematical and physical worlds.