The 2023 Distinct Distances Project
This is a brief introduction to the 2023 Polymath Jr project that will be run by Eyvi Palsson and Adam Sheffer. It concerns the famous distinct distances problem of Erdős.
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Consider this set of five points:
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Every pair of points span a distance. When checking all pairs, we get that the distances spanned by this set are {√2, 2, √5, √10}. This is the set of distinct distances. It does not matter how many times each distance appears - only that it appears at least once.
The five vertices of a regular pentagon only span two distances:
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Erdős was interested in much larger sets of points. For a huge n, he asked for the minimum number of distinct distances that can be spanned by n points. For example, n equally spaced points on a line span exactly n-1 distances.
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Erdős arranged n points so that they spanned about n/√(log n) points. He conjectured that this is the best possible. This has led to a huge body of work, spanning many decades, and deep mathematical theories.
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In the current Polymath Jr project, we return to small sets of points. Let g(k) be the maximum number of points that span exactly k distances. For example:
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g(1)=3 since the three vertices of an equilateral triangle span only one distance. A set of four points cannot span only one distance.
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g(2)=5, as demonstrated by the above regular pentagon. A set of six points cannot span only two distances.
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Three different ways for 9 points to span only four distances:
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In this project, we will continue to explore the g(k) problem. A few potential approaches:
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The largest value that is currently known is g(6)=13. We could work towards proving g(7).
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Participants could search for larger configurations that span few distances, either by hand or using a computer.
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Derive more general bounds for every k.
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More experienced participants could try applying more modern techniques to this problem (we have some specific ideas for such techniques).
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Anything else - go wild!
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For additional reading, see:​
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We will provide a lot more material, but for that you'll have to join the project!
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