On the zeros of cosine polynomials: solution to a problem of Littlewood


Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [12, Problem 22] poses the following research problem, which appears to be still open:

Problem. “If the $n_j$ are integral and all different, what is the lower bound on the number of real zeros of $\sum_{j=1}^N \cos (n_j\theta)$? Possibly $N-1$, or not much less.”

No progress seems to have been made on this in the last half century. We show that this is false.


Peter Borwein

Department of Mathematics
Simon Fraser University
Burnaby BC V5A 1S6

Tamás Erdélyi

Department of Mathematics, Texas A&M University, College Station, TX 77843, United States

Ron Ferguson

Department of Mathematics, Simon Fraser University, Burnaby BC V5A 1S6, Canada

Richard Lockhart

Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby BC V5A 1S6, Canada