LOWER BOUNDS FOR THE NUMBER OF ZEROS OF COSINE POLYNOMIALS IN THE PERIOD: A PROBLEM OF LITTLEWOOD Peter Borwein and Tam´ as Erd´ elyi Abstract. Littlewood in his 1968 monograph “Some Problems in Real and Complex Anal- ysis” [9, problem 22] poses the following research problem, which appears to still be open: “If the n m are integral and all different, what is the lower bound on the number of real zeros of P N m=1 cos(n m θ)? Possibly N − 1, or not much less.” Here real zeros are counted in a period. In fact no progress appears to have been made on this in the last half century. In a recent paper [2] we showed that this is false. There exists a cosine polynomial P N m=1 cos(n m θ) with the n j integral and all different so that the number of its real zeros in the period is O(N 9/10 (log N ) 1/5 ) (here the frequencies n m = n m (N ) may vary with N ). However, there are reasons to believe that a cosine polynomial P N m=1 cos(n m θ) always has many zeros on the period. Denote the number of zeros of a trigonometric polynomial T in the period [−π,π) by N (T ). In this paper we prove the following. Theorem. Suppose the set {a j : j ∈ N}⊂ R is finite and the set {j ∈ N : a j =0} is infinite. Let T n (t)= n X j=0 a j cos(jt) . Then lim n→∞ N (T n )= ∞ . One of our main tools, not surprisingly, is the resolution of the Littlewood Conjecture [4]. 1. Introduction Let 0 ≤ n 1 <n 2 < ··· <n N be integers. A cosine polynomial of the form T N (θ)= ∑ N j =1 cos(n j θ) must have at least one real zero in a period. This is obvious if n 1 = 0, since then the integral of the sum on a period is 0. The above statement is less obvious if n 1 = 0, but for sufficiently large N it follows from Littlewood’s Conjecture simply. Here we mean the Littlewood’s Conjecture proved by S. Konyagin [5] and independently by McGehee, Pigno, and Smith [11] in 1981. See also [4] for a book proof. It is not difficult to prove the statement in general even in the case n 1 = 0. One possible way is to use the identity n N  j =1 T N ((2j − 1)π/n N )=0 . Key words and phrases. a problem of Littlewood, cosine polynomials, constrained coefficients, number of real zeros. 2000 Mathematics Subject Classifications: Primary: 41A17 Typeset by A M S-T E X 1