# A conjecture of Littlewood

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

• Do estimates on the constant $C$ exist? – lcv Oct 8 '18 at 17:43
• Glancing at Konyagin's paper( it's in Russian and I worked so hard to follow his proof) its stated that $C\leq 1$. But again its been over three decades and I'm pretty sure better estimates are out there – BigM Oct 12 '18 at 21:01