theoremnovelinferred_dep

Power-saving bound for planar unit distances

Theorem·thm:main·§1 Introduction

Statement (paper)

There are absolute constants 0<C<∞0<C<\infty and 1≤β<4/31\le\beta<4/3 such that u(n)≤Cnβu(n)\le Cn^\beta for every nonnegative integer nn.

In plain English

There exist absolute constants CC and β<4/3\beta<4/3 (with β≥1\beta\ge1) such that the maximum number of unit distances among nn points in the plane satisfies u(n)≤Cnβu(n)\le Cn^\beta for all nn.

Why it's here

This is the paper's main result, improving the classical Spencer–Szemerédi–Trotter exponent 4/34/3 by a power. It follows by contradiction: an extremal configuration with n4/3−o(1)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.

Uses

propTotal variation S cannot stay boundedpropOrganized pair graph with height-controlled ratiospropNo algebraic configuration with large heightspropIncidence piece with degree and expansion control

Used by

Nothing in the paper

Sources

PaperOpen
Section
§1 Introduction
Page
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Label
thm:main
Line
sections/00-introduction.tex:12
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    "the sequence reduction in " giving configurations with t^{4-o(1)} ordered unit pairs, and "the incidence pieces used throughout the proof"