theoremnovelinferred_dep

Power-saving bound for planar unit distances

Theorem·thm:main·§1 Introduction

Statement (paper)

There are absolute constants

0<C<\infty

and

1\le\beta<4/3

such that

u(n)\le Cn^\beta

for every nonnegative integer

n

In plain English

There exist absolute constants

C

and

\beta<4/3

(with

\beta\ge1

) such that the maximum number of unit distances among

n

points in the plane satisfies

u(n)\le Cn^\beta

for all

n

Why it's here

This is the paper's main result, improving the classical Spencer–Szemerédi–Trotter exponent

4/3

by a power. It follows by contradiction: an extremal configuration with

n^{4/3-o(1)}

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.

Selected node

Statement (paper)

In plain English

Why it's here