Littlewood's L1 norm conjecture

Remarks

Littlewood conjectured this bound, and it was proved in 1981, independently by McGehee, Pigno and Smith and by Konyagin. McGehee, Pigno and Smith proved more: for all complex $a_1, \ldots, a_N$, $$\frac{1}{2\pi} \int_{-\pi}^{\pi} \Big\lvert \sum_{k=1}^{N} a_k e^{i n_k x} \Big\rvert \, dx \ge c \sum_{k=1}^{N} \frac{\abs{a_k}}{k}.$$ Taking all $a_k = 1$ gives the conjecture, because $\sum_{k=1}^{N} 1/k \ge \log N$. Their proof constructs, by duality, a bounded function whose pairing with $f$ is large.

The order $\log N$ is sharp. For $n_k = k$ one has $\abs{f(x)} = \abs{\sin(Nx/2) / \sin(x/2)}$, the absolute value of a Dirichlet-type kernel, and its $L^1$ norm grows like $\frac{4}{\pi^2} \log N$, as for the Lebesgue constants.

Going deeper

For a student

In the $L^2$ norm the answer is exact: the norm is $\sqrt N$ by Parseval's identity, whatever the frequencies. The $L^1$ norm is much harder because no orthogonality is available. The McGehee–Pigno–Smith inequality is a version of Hardy's inequality for functions whose Fourier coefficients are supported on an arbitrary set of integers. Before the full solution, P. J. Cohen proved a lower bound tending to infinity with $N$, as part of his work on idempotent measures.