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Roth's theorem on arithmetic progressions
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<p>Roth's theorem on arithmetic progressions is a result in <a href="/facts/Additive_combinatorics/wfvGQVUB">additive combinatorics</a> concerning the existence of <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progressions</a> in subsets of the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>. It was first <a href="/facts/Mathematical_proof/7ECDrU80">proven</a> by <a href="/facts/Klaus_Roth/exBd7vDU">Klaus Roth</a> in 1953. Roth's theorem is a special case of <a href="/facts/Szemer%C3%A9di%27s_theorem/6fjvXcEx">Szemerédi's theorem</a> for the case k = 3 {\displaystyle k=3} . </p>