OpenAI’s diagram is based on choosing c² = 65, which can be satisfied by 1² + 8² = 65 or 4² + 7² = 65. This means that if the grid spacing is 1/√65, each point will be one unit away from 16 other points: (1,8), (4,7), (7,4), (8,1), (-1,8), (-4,7), and so on. Larger values for C² – if they are chosen carefully – enable more whole-number diagonals and hence more unit-distance pairs.
However, if c² is very large compared to the number of points in the grid, many of the possible one-unit-away neighbors will be outside the grid.
In short, we want to choose a c² that is large enough but not too large. Using insights from number theory, including Jacobi’s two-square theorem, Erdős was able to show that an optimally sized circle would enable the number of unit-distance pairs to grow faster than the number of points, but only barely.
The question became “Can you do better?” To find the upper bound, Erdős used an argument from an entirely different area of mathematics, called graph theory, to show that you can only have so many unit distances. But his upper bound grows much faster than the best lower bound he was able to create.
Erdős estimated that the true optimum was much closer to the lower bound than the upper bound. He predicted, but could not prove, that the maximum number of unit-distance pairs grows barely faster than the number of points.
To be more precise, Erdös estimated the number of unit distances to be n^(1+o(1)). In other words, for sufficiently large n, the maximum number of unit distances will be less than n^(1+𝜖) for any 𝜖 > 0. This may grow slightly faster than his lower-bound construction – which was n^(1 + C/(log log n)) for some constant C – but within the same general ballpark.
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