Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then $$\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,$$ where $e(x)=e^{2\pi ix }$?

Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then $$\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,$$ where $e(x)=e^{2\pi ix }$?

Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81].