Littlewood's Cosine Root Problem

Problem's Description

Let $A \subset \mathbb{N}$ be a set of positive integers. One basic intuition is that the function

$$ f(x) = \sum_{k \in A} \cos(kx)$$

should have many roots on $[0, 2\pi]$ when the set $A$ is large. Littlewood conjectured that the number of roots is probably something like $\#A - 1$. This was disproven in 2008 by Borwein, Erdelyi, Ferguson, Lockhart (Annals). The best bounds are

$$ (\log \log \log \# A)^{1-} \lesssim \mbox{number of roots} \lesssim (\# A \log \# A)^{2/3},$$

where the lower bound is due to Erdelyi (2017) and the upper bound is due to Juskevicius-Sahasrabudhe (2020).

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

   • P. Borwein,
   • T. Erdelyi,
   • R. Ferguson,
   • R. Lockhart

   Annals of Mathematics

2. Article Counting zeros of cosine polynomials: on a problem of Littlewood.

   • J. Sahasrabudhe

   Advances in MathematicsarXiv

3. Article Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

   • T. Erdelyi

   arXiv

4. Article Cosine polynomials with few zeroes

   • T Juškevičius,
   • J Sahasrabudhe

   arXiv