Benjamin Bedert

Zeros of polynomials with restricted coefficients and a problem of Littlewood

It is an old problem of Littlewood to estimate the minimum number $Z(N)$ of zeros in $[0,2\pi]$ that a function of the form $f_A(x)=\sum_{n\in A}\cos(nx)$ can have, where $A$ is a set of integers of size $N$. We discuss recent progress showing that $Z(N)\gg (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous bound $(\log\log\log N)^c$ due to Erd\'elyi and Sahasrabudhe. We also consider a closely related question due to Borwein, Erd\'elyi and Littmann which asks about the minimum number of zeros $Z_{\mathcal{L}}(N)$ of a cosine polynomials with $\pm 1$-coefficients. Until recently even the qualitative behaviour, i.e. whether $Z_{\mathcal{L}}(N)$ tends to infinity with the degree $N$, was unknown and a conjecture due to Drungilas, Erd\'elyi, and Mukunda. We also discuss our work confirming this conjecture.

This talk is based on the papers `An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials' and `On the zeros of reciprocal Littlewood polynomials'.