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  "body": "\n## Summary\n\nAdditive combinatorics studies finite subsets of abelian groups, such as sets of integers, and how they behave under addition.\nTypical questions ask how large a [[sumset|XNBN31HQWB]] $A + B$ must be, and what a set with a small sumset looks like.\nOthers ask when a set contains additive patterns such as arithmetic progressions.\nIt combines combinatorics, Fourier analysis, ergodic theory and number theory.\nIts results include [[Szemerédi's theorem|B23B0QEMN5]], [[Freiman's theorem|KEPFYE6WYZ]] and the [[Green–Tao theorem|H8BAHWSEMC]].\n\n## Explanations\n\n### L1 human\n\nTake a finite set of numbers and add every pair of them, allowing a number to be added to itself.\nAdditive combinatorics asks how many different sums you get, and what that number tells you about the set.\nFor example, $\\set{1, 2, 3, 4}$ gives only the sums $2, \\dots, 8$, seven values, because it is very structured.\nA random-looking set like $\\set{1, 2, 5, 11}$ gives ten different sums.\nThe field also asks when a large set must contain evenly spaced numbers.\n",
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