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  "body": "\n## Definition\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$ be the circle.\nA subset $E \\subset \\mathbb{T}$ is a *set of uniqueness* if the following holds.\nWhenever a [[trigonometric series|FZK2M8KGXQ]] $\\sum_{n \\in \\Z} c_n e^{inx}$ with complex coefficients satisfies\n$$\\lim_{N \\to \\infty} \\sum_{n=-N}^{N} c_n e^{inx} = 0 \\quad \\text{for every } x \\in \\mathbb{T} \\setminus E,$$\nthen $c_n = 0$ for all $n \\in \\Z$.\nA subset of $\\mathbb{T}$ that is not a set of uniqueness is a *set of multiplicity*.\n\n## Remarks\n\nSets of uniqueness are often called U-sets, and sets of multiplicity M-sets.\nA subset of a set of uniqueness is again a set of uniqueness.\n\nExample: the empty set is a set of uniqueness; this is [[Cantor's uniqueness theorem|NSS3XW4BYZ]].\nEvery [[countable set|01TQB88H4C]] is also a set of uniqueness, a result of W. H. Young.\n\nNon-example: let $E$ contain an open arc $I$ of $\\mathbb{T}$.\nChoose a nonzero infinitely differentiable function $g$ on $\\mathbb{T}$ that vanishes outside $I$.\nIts Fourier series converges absolutely to $g$ at every point, so it converges to $0$ outside $E$.\nIts coefficients are not all $0$, because $g \\ne 0$, so $E$ is a set of multiplicity.\n\nMore generally every set of positive Lebesgue measure is a set of multiplicity, and a Borel set of uniqueness has measure zero.\nThe converse fails: [[Menshov's example|AW16MYJV5Q]] is a closed set of measure zero that is a set of multiplicity.\nThe middle-thirds [[Cantor set|0BXNDZFQME]] is a set of uniqueness.\n",
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