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 found here