theoremnovelinferred_dep
Power-saving bound for planar unit distances
Theorem·
thm:main·§1 IntroductionStatement (paper)
There are absolute constants and such that
for every nonnegative integer .
In plain English
There exist absolute constants and (with ) such that the maximum number of unit distances among points in the plane satisfies for all .
Why it's here
This is the paper's main result, improving the classical Spencer–Szemerédi–Trotter exponent by a power. It follows by contradiction: an extremal configuration with unit distances would yield the incidence piece, which rules out bounded total variation and gives the height-controlled pair graph, and that graph is then excluded by the algebraic obstruction.
Classified novel:
This is the paper's headline result. It is the first power-saving improvement over the Spencer–Szemerédi–Trotter bound u(n)=O(n^{4/3}) (1984), which had stood as the best known exponent for the Erdős unit distance problem.