LITTLEWOOD-TYPE PROBLEMS ON SUBARCS OF THE UNIT-CIRCLE

The results of this paper show that many types of polynomials cannot b e small on subarcs of the unit circle in the complex plane. A typical result of the paper is the following. Let F-n denote the set of polyno mials of degree at most n with coefficients from {-1,0,1}. There are a bsolute constants c(1) > 0, c(2) > 0, and c(3) > 0 such that [GRAPHICS ] for every subarc A of the unit circle partial derivative D := {z is an element of C : \z\ = 1} with length 0 < a < c(3). The lower bound r esults extend to the class of f of the form [GRAPHICS] with any nonneg ative integer m less than or equal to 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 for m that is arbitrarily small on as much of the boundary (in the sense o f linear Lebesgue measure) as one likes. An easy to formulate corollar y of the results of this paper is the following. Corollary. Let A be a subarc of the unit circle with length l(A) = a. If (p(k)) is a sequen ce of monic polynomials that tends to 0 in L-1(A), then the sequence ( H(p(k))) of heights tends to infinity. The results of this paper are d ealing with (extensions of) classes much studied by Littlewood and man y others in regards to the various conjectures of Littlewood concernin g growth and flatness of unimodular polynomials on the unit circle do. Hence the title of the paper.