Pinned distances and a power saving for planar unit distances at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Pinned distances and a power saving for planar unit distances

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	2
Category:	Combinatorics	Lean version:	YES! ✔
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Planar distinct distances and unit-distance bounds. Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $o(n)$ points of any n-point planar set determine at least $n^{1-\varepsilon}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\delta})$ for an absolute δ > 0.

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LEVEL 1 The weak pinned planar distance theorem

released 2026-09-23 | 1 theorem · 7 lemmas · 9 proofs · 10,922 words | PLAY LEVEL 1 » (pdf)

We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but $o(n)$ points of any n-point planar set determine at least $n^{1-\varepsilon}$ distinct nonzero distances.

LEVEL 2 A power saving for planar unit distances

released 2026-09-23 | 1 theorem · 27 lemmas · 38 proofs · 23,123 words | PLAY LEVEL 2 » (pdf)

We prove a power saving for the planar unit-distance problem: for some absolute $\beta\lt 4/3$, every set of n points in the Euclidean plane determines $O(n^\beta)$ unordered pairs at unit distance.

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