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  "body": "\n## Summary\n\nThe Schnirelmann density of a set $A$ of positive integers is the infimum over $n \\ge 1$ of the proportion of $\\set{1, \\dots, n}$ that lies in $A$.\nUnlike the [[natural density|RVG9Z8NYA6]], it always exists and it is very sensitive to small elements: it is $0$ whenever $1 \\notin A$.\nIt behaves well under sums of sets, which makes it a tool for additive problems.\nSchnirelmann used it to show that every integer greater than $1$ is a sum of a bounded number of [[primes|01826KMG2H]].\n\n## Explanations\n\n### L1 human\n\nLook at the first $n$ positive integers and ask what fraction of them belongs to $A$.\nThe Schnirelmann density is the worst such fraction over all $n$.\nFor the odd numbers the fraction is $1$ at $n = 1$ and $1/2$ at every even $n$, so the density is $1/2$.\nFor the even numbers it is already $0$ at $n = 1$, so the density is $0$.\nA set with positive Schnirelmann density has to start early and keep up its pace everywhere.\n",
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