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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].