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  "body": "\n## Definition\n\nLet $A$ be a subset of the [[natural numbers|0MWKJ44V5A]].\nFor each integer $n \\ge 1$, let $A(n)$ be the number of elements $a \\in A$ with $1 \\le a \\le n$.\nThe *Schnirelmann density* of $A$ is\n$$\\sigma(A) = \\inf_{n \\ge 1} \\frac{A(n)}{n}.$$\n\n## Remarks\n\nAlways $0 \\le \\sigma(A) \\le 1$, and $A(n) \\ge \\sigma(A)\\, n$ for every $n \\ge 1$.\nThe element $0$, if present, does not count.\nOne has $\\sigma(A) = 1$ if and only if $A$ contains every positive integer.\n\nExample: for the set $A$ of odd positive integers, $A(n)/n = 1/2$ for even $n$ and $A(n)/n = (n+1)/(2n) \u003e 1/2$ for odd $n$; so $\\sigma(A) = 1/2$.\nContrast: for the even positive integers, $A(1)/1 = 0$, so $\\sigma(A) = 0$, although their [[natural density|RVG9Z8NYA6]] is $1/2$.\n\nSchnirelmann's inequality: if $0 \\in A$ and $0 \\in B$, then $\\sigma(A + B) \\ge \\sigma(A) + \\sigma(B) - \\sigma(A)\\sigma(B)$, where $A + B = \\set{a + b : a \\in A, b \\in B}$.\n",
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