Littlewood-type problems on subarcs of the unit circle

Abstract

The results of this paper show that many types of polynomials cannot be small on subarcs of the unit circle in the complex plane. A typical result of the paper is the following. Let Fn denote the set of polynomials of degree at most n with coefficients from {-1,0,1}. There are absolute constants c1 > 0, c2 > 0, and c3, > 0 such that exp(-c1/a) ≤ inf ∥p∥L1(A), and 0≠p∈Fn inf ∥p∥A ≤ exp (-c2/a) 0≠p∈Fn for every subarc A of the unit circle ∂D := {z ∈ ℂ : \z| =1} with length 0 < a < c3. The lower bound results extend to the class of f of the form f(z) = nΣj=m ajz0, aj ∈ ℂ, |aj| ≤ M, |am| = 1 with any nonnegative integer m < n. It is shown that functions f of the above form cannot be arbitrarily small uniformly on subarcs of the circle. However, this does not extend to sets of positive measure. It is shown that it is possible to find a polynomial of the above form that is arbitrarily small on as much of the boundary (in the sense of linear Lebesgue measure) as one likes. An easy to formulate corollary of the results of this paper is the following.