On the minimum modulus of trigonometric polynomials

For all even integers N greater than 2, a trigonometric polynomial fN(X) = Zk=-N akeikx satisfying lakl 0. This problem is part of the more general question, as expressed by Beller [Be]: "How close can we get to a situation where P(z) is a polynomial of degree n > 0 which, on the one hand, has coefficients of constant modulus, and on the other hand IP(z)I is constant for IzI = 1 ?" The question goes back to Littlewood [L] and has also been studied by Beller and Newman [BeN], Byrnes [By], Korner [Ko], and Kahane [Ka]. Problem 27 of Erdos [E] is also concerned with this topic. From the standpoint of existence, the question has been emphatically answered by Kahane [Ka] where he proves the following. Theorem. There exists a sequence of polynomials n Pn(z) = E amn z m=1 with lam, n I = 1 and a positive sequence en converging to 0 such that for all Z I = 1 the following holds: ( I18 n)> < I|Pn (z) I < (1+ 8n) Vn However, Kahane's proof is probabilistic and it is not known how to construct such polynomials. It is not even known how to construct polynomials which satisfy A vH < IPn (z) I < B vi for some pair of absolute constants 0 < A < B < oc. Received by the editors September 24, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 42A05.