The following is a conjecture due to Littlewood.

> For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality
> $$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$  holds.

Has this proven to be true or false?

**Update 1.** An extension to finite fields can be find [here][1]


  [1]: https://www.researchgate.net/profile/Victor_Garcia11/publication/327356917_The_finite_Littlewood_problem_in_mathbb_F_pFp/links/5b969d5792851c78c410183b/The-finite-Littlewood-problem-in-mathbb-F-pFp.pdf?origin=publication_detail