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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\Lambda \\subset \\R^n$ be a [[lattice|WN9V7WSJKF]] with covolume $\\det \\Lambda$.\nLet $S \\subseteq \\R^n$ be a [[Lebesgue measurable|2Z2QVWZ262]] set with volume $\\operatorname{vol}(S) \u003e \\det \\Lambda$.\nThen there are two distinct points $x, y \\in S$ such that $x - y \\in \\Lambda$.\n\n## Remarks\n\nLet $P$ be a fundamental parallelepiped of $\\Lambda$, so that the translates $P + v$ with $v \\in \\Lambda$ are pairwise disjoint and cover $\\R^n$.\nCut $S$ into the pieces $S \\cap (P + v)$ and translate each piece by $-v$ into $P$.\nThe translated pieces have total volume $\\operatorname{vol}(S) \u003e \\operatorname{vol}(P) = \\det \\Lambda$, so two of them overlap; a common point gives $x - v = y - w$ with $x \\ne y$ in $S$ and $v \\ne w$ in $\\Lambda$.\n[[Minkowski's theorem|0K387S2PKQ]] follows: if $K$ is convex, symmetric about $0$, and $\\operatorname{vol}(K) \u003e 2^n \\det \\Lambda$, apply the theorem to $S = \\tfrac12 K$.\nIt gives $x \\ne y$ in $S$, and $x - y = \\tfrac12(2x) + \\tfrac12(-2y)$ is a nonzero point of $K \\cap \\Lambda$.\nThe bound is sharp: the half-open square $[0, 1)^2$ has area $1 = \\det \\Z^2$, and no two of its distinct points differ by an integer vector.\nMore generally, if $\\operatorname{vol}(S) \u003e k \\det \\Lambda$ for a positive integer $k$, the same argument gives $k + 1$ distinct points of $S$ whose pairwise differences lie in $\\Lambda$.\n\n## Explanations\n\n### L1 human\n\nMove every point of a region of the plane by whole steps left, right, up or down until it lands in the unit square.\nIf the region has area larger than $1$, the moved pieces cannot all fit in the square without overlapping.\nSo the region contains two points that differ by a whole-number shift.\nThe theorem says the same for any lattice in any dimension, with the volume of a lattice cell in place of $1$.\n\n### L2 human\n\nBlichfeldt's theorem is the measure-theoretic pigeonhole principle behind the geometry of numbers.\nIt gives a short proof of Minkowski's convex body theorem, and through it of results such as the two-square and four-square theorems.\nIt holds for any measurable set, convex or not, which makes it useful when the region of interest is not convex.\n",
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