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  "body": "\n## Problem\n\nLet $n \\ge 2$ be an integer.\nFor a set $S$ of $n$ points in the plane $\\R^2$, consider the [[Euclidean distances|0NPPGHQNXQ]] $\\norm{p - q}$ with $p, q \\in S$ and $p \\ne q$.\nLet $g(n)$ be the least number of distinct values of these distances, over all such sets $S$.\nFind the order of growth of $g(n)$ as $n \\to \\infty$.\nErdős conjectured that the square grid is essentially optimal.\nA weaker form of his conjecture says that $g(n) \\ge n^{1 - \\varepsilon}$ for every $\\varepsilon \u003e 0$ and all large enough $n$.\n\n## Remarks\n\nErdős posed the problem in 1946 and proved the lower bound $g(n) = \\Omega(\\sqrt{n})$.\nA $\\sqrt{n} \\times \\sqrt{n}$ piece of the integer lattice determines only $O(n / \\sqrt{\\log n})$ distinct distances, so $g(n) = O(n / \\sqrt{\\log n})$.\nGuth and Katz proved the lower bound $g(n) = \\Omega(n / \\log n)$.\nThis settles the weaker conjecture $g(n) \\ge n^{1 - \\varepsilon}$.\nThe exact order of $g(n)$ is still open: the known bounds differ by a factor of order $\\sqrt{\\log n}$.\nThe analogous questions in higher dimensions are also studied.\n\n## Explanations\n\n### L1 human\n\nPut $n$ dots in the plane and list all the distances between pairs of dots.\nHow few different values can this list contain?\nIf the dots are spread at random, almost all distances differ.\nA square grid does much better, because many pairs of dots have the same distance.\nThe problem asks whether one can do much better than a grid, and the answer is essentially no.\n",
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