Journal of the
Korean Mathematical Society JKMS

Let $\zeta$ be a fixed complex number. In this paper, we study the quantity $S(\zeta,n):=\max_{f \in \Lambda_n} |f(\zeta)|,$ where $\Lambda_n$ is the set of all real polynomials of degree at most $n-1$ with coefficients in the interval $[0,1].$ We first show how, in principle, for any given $\zeta \in \mathbb C$ and $n \in \mathbb N,$ the quantity $S(\zeta,n)$ can be calculated. Then we compute the limit $\lim_{n \to \infty} S(\zeta,n)/n$ for every $\zeta \in \mathbb C$ of modulus $1.$ It is equal to $1/\pi$ if $\zeta$ is not a root of unity. If $\zeta=\exp(2\pi i k/d),$ where $d \in \mathbb N$ and $k \in [1,d-1]$ is an integer satisfying gcd$(k,d)=1,$ then the answer depends on the parity of $d.$ More precisely, the limit is $1,$ $1/(d\sin(\pi/d))$ and $1/(2d\sin(\pi/2d))$ for $d=1,$ $d$ even and $d>1$ odd, respectively.

Keywords: Newman polynomial, maximum of a polynomial, root of unity, Dirichlet's theorem

MSC numbers: 26C05, 12D10, 12E10, 11R09, 11J04