Breaking Erdős: A 1.014 Exponent for Unit Distances
This lightning talk explores a breakthrough in discrete geometry that refutes a decades-old conjecture by Paul Erdős. Using sophisticated tools from algebraic number theory, the paper constructs explicit planar point sets of size n containing more than n^1.014 pairs at unit distance—a polynomial improvement over the conjectured near-linear bound. We'll unpack how CM fields, lattice projections, and class group bounds combine to achieve this striking result, and examine what it means for the future of geometric extremal problems.Script
For decades, mathematicians believed that any collection of points in the plane could have at most a near-linear number of pairs at exactly unit distance apart. This paper shatters that intuition with an explicit construction achieving n to the 1.014 pairs for n points.
Erdős himself gave a lower bound that grew just barely faster than linear, and he conjectured that was essentially optimal. A recent computer-assisted proof hinted at a tiny polynomial improvement, but the exponent was microscopically small and not explicit.
The construction begins in high dimensions, using number fields called CM fields to build lattices where many vectors have carefully controlled norms. When these lattices are projected down to the plane, those special vectors become pairs of points at exactly unit distance.
The technical heart is an infinite tower of number fields with exquisitely controlled properties. By applying the Golod-Shafarevich criterion and explicit class group bounds, the authors ensure that ideals split in just the right way to maximize unit distance pairs.
The method cannot push beyond n to the 1.243 even under the most optimistic conditions, so there's a ceiling. But the achieved exponent of 1.014 is fully explicit, with every constant and parameter computed, making this the first quantitative polynomial improvement over Erdős's bound.
This result proves that deep algebraic machinery can overturn long-standing combinatorial conjectures, and it opens doors to attacking other extremal problems where coordinates carry hidden structure. To dive deeper into this breakthrough and create your own explainer videos, visit EmergentMind.com.
