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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f \\in L^1(\\R^n)$.\nLet $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ be its [[Fourier transform|DPVM34ADZV]].\nSuppose that the sets $\\set{x \\in \\R^n : f(x) \\ne 0}$ and $\\set{\\xi \\in \\R^n : \\hat f(\\xi) \\ne 0}$ both have finite [[Lebesgue measure|2Z2QVWZ262]].\nThen $f = 0$ almost everywhere.\n\n## Remarks\n\nThe theorem strengthens the [[qualitative uncertainty principle|S1FB2K14NV]], which assumes bounded sets instead of sets of finite measure.\nWith finite measure only, the analytic continuation argument is no longer available.\nThe proof periodizes instead.\nAfter a dilation one may assume that the set where $f \\ne 0$ has measure less than $1$.\nFor almost every $\\xi$, the $\\Z^n$-periodization of $x \\mapsto f(x) e^{-2\\pi i x \\cdot \\xi}$ has the numbers $\\hat f(\\xi + k)$, $k \\in \\Z^n$, as Fourier coefficients, as in the [[Poisson summation formula|GARGJYSGFG]], and only finitely many of them are nonzero.\nSo this periodization is a trigonometric polynomial, and it vanishes on a set of positive measure in the torus.\nA trigonometric polynomial that vanishes on a set of positive measure is zero, so $\\hat f(\\xi + k) = 0$ for all $k$ and almost every $\\xi$, and $f = 0$.\n\nThe theorem is due to M. Benedicks.\nA version for $L^2(\\R^n)$ was proved independently by W. O. Amrein and A. M. Berthier.\n\nExample: the Gaussian $e^{-\\pi \\abs{x}^2}$ is nowhere zero, as is its transform, so both sets have infinite measure, consistent with the theorem.\n\n## Explanations\n\n### L1 human\n\nA nonzero function cannot live on a set of finite total size while its spectrum also lives on a set of finite total size.\nThe sets may be scattered and unbounded; only their total size matters.\n\n### L2 human\n\nThis is the measure-theoretic form of the principle that $f$ and $\\hat f$ cannot both be concentrated.\nA quantitative version, due to Nazarov in dimension one and extended to $\\R^n$, bounds $\\norm{f}_2$ by a constant times the $L^2$ norms of $f$ and $\\hat f$ outside such sets.\nA pair of sets with this property is called an annihilating pair, and the theorem says that any two sets of finite measure form one.\n",
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