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  "body": "\n## Statement\n\nLet $A$ be a set of positive integers with\n$$\\limsup_{N \\to \\infty} \\frac{\\abs{A \\cap \\set{1, \\dots, N}}}{N} \u003e 0,$$\nwhere $\\limsup$ is the [[limit superior|0T3RZQXDYD]].\nThen $A$ contains three terms $a, a + d, a + 2d$ of an arithmetic progression with $d \\ge 1$.\n\n## Remarks\n\nLet $r_3(N)$ be the largest size of a subset of $\\set{1, \\dots, N}$ with no three-term arithmetic progression.\nThe theorem is equivalent to $r_3(N) / N \\to 0$ as $N \\to \\infty$.\nRoth's proof uses Fourier analysis and a density increment.\nA set without three-term progressions has a large Fourier coefficient, so it is denser on some long sub-progression.\nThis step cannot repeat forever, because the density cannot exceed $1$.\nAnother proof goes through the [[triangle removal lemma|XMS1QRAQZC]].\nBehrend constructed sets showing $r_3(N) \\ge N \\exp(-c\\sqrt{\\log N})$ for a constant $c \u003e 0$, so $r_3(N)$ is not bounded by $N^{1 - \\varepsilon}$ for any $\\varepsilon \u003e 0$.\nKelley and Meka proved in 2023 an upper bound of the form $r_3(N) \\le N \\exp(-c (\\log N)^{\\beta})$ with constants $c, \\beta \u003e 0$.\nThe analogue in $\\F_3^n$ is the [[cap set problem|1SF8Y9M8YB]].\n\n## Explanations\n\n### L1 human\n\nTake any set of whole numbers that fills a fixed positive fraction of the numbers.\nThen it contains three numbers equally spaced, like $5, 9, 13$.\nThe odd numbers, for example, contain $1, 3, 5$.\n\n### L2 human\n\nRoth's theorem is the case $k = 3$ of [[Szemerédi's theorem|B23B0QEMN5]], and it was proved first.\nIts Fourier-analytic density-increment argument is a model for many proofs in [[additive combinatorics|3NM362A2JG]].\nThe gap between Behrend's lower bound and the best upper bounds for $r_3(N)$ is still being studied.\n",
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