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  "body": "\n## Statement\n\nThere is an absolute constant $c \u003e 0$ with the following property.\nLet $N \\ge 1$ and let $n_1 \u003c n_2 \u003c \\cdots \u003c n_N$ be distinct integers.\nThen the [[trigonometric polynomial|J3NJQ5FZJ5]] $f(x) = \\sum_{k=1}^{N} e^{i n_k x}$ satisfies\n$$\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{f(x)} \\, dx \\ge c \\log N.$$\n\n## Remarks\n\nLittlewood conjectured this bound, and it was proved in 1981, independently by McGehee, Pigno and Smith and by Konyagin.\nMcGehee, Pigno and Smith proved more: for all complex $a_1, \\ldots, a_N$,\n$$\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\Big\\lvert \\sum_{k=1}^{N} a_k e^{i n_k x} \\Big\\rvert \\, dx \\ge c \\sum_{k=1}^{N} \\frac{\\abs{a_k}}{k}.$$\nTaking all $a_k = 1$ gives the conjecture, because $\\sum_{k=1}^{N} 1/k \\ge \\log N$.\nTheir proof constructs, by duality, a bounded function whose pairing with $f$ is large.\n\nThe order $\\log N$ is sharp.\nFor $n_k = k$ 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|RWKQ5EP4YF]].\n\n## Explanations\n\n### L1 human\n\nAdd $N$ pure waves of size one with different whole-number frequencies.\nAt $x = 0$ they all line up and the sum has size $N$, but at most points they partly cancel.\nThe theorem says that the cancellation can never be too strong on average: the average size of the sum is at least a constant times $\\log N$.\n\n### L2 human\n\nIn the $L^2$ norm the answer is exact: the norm is $\\sqrt N$ by Parseval's identity, whatever the frequencies.\nThe $L^1$ norm is much harder because no orthogonality is available.\nThe McGehee–Pigno–Smith inequality is a version of Hardy's inequality for functions whose Fourier coefficients are supported on an arbitrary set of integers.\nBefore the full solution, P. J. Cohen proved a lower bound tending to infinity with $N$, as part of his work on idempotent measures.\n",
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