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  "body": "\n## Summary\n\nA Sidon set is a set of integers, or of elements of an abelian group, in which all sums of two elements are different.\nHere sums that differ only in the order of the summands count as the same.\nFor sets of integers this is the same as saying that all differences of two distinct elements are different.\nSidon sets are the extreme opposite of sets with additive structure: their [[additive energy|MAQKH72P6V]] is as small as possible.\nThe largest Sidon subset of $\\set{1, \\dots, N}$ has size asymptotic to $\\sqrt{N}$.\n\n## Explanations\n\n### L1 human\n\nPick numbers so that no two pairs of them have the same sum.\nThe powers of two $1, 2, 4, 8, 16$ work, because binary expansions show each sum of two of them only once.\nSuch sets cannot be very dense: among the numbers up to $N$, a Sidon set has only on the order of $\\sqrt{N}$ elements.\nThe reason is that its roughly $n^2/2$ pairwise sums must all be different and at most $2N$.\n",
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