Bounds on the exceptional set in the $abc$ conjecture

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 $\hat{I}\mu >0$, such triples satisfy $\mathrm{rad}(abc)\ge c^{1-\hat{I}\mu }$ 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.

12 pages; Python linear programming code included in the appendix