Bounds on the exceptional set in the $abc$ conjecture

TL;DR

This paper establishes a power-saving bound on the size of the exceptional set of coprime triples related to the $abc$ conjecture, using advanced techniques from geometry of numbers and Fourier analysis.

Contribution

It provides the first explicit power-saving bound on the exceptional set in the $abc$ conjecture, advancing understanding of the conjecture's finiteness aspect.

Findings

Derived a power-saving bound on the exceptional set size

Applied geometry of numbers to analyze integer points on varieties

Utilized Fourier analysis techniques in the proof

Abstract

We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many exceptions. In this article we obtain a power-saving bound on the size of the exceptional set of triples. The proof is based on a combination of upper bounds for the density of integer points on certain high-dimensional varieties, coming from the geometry of numbers and from Fourier analysis.