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 $ε>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-ε}$ with finitely many exceptions. In this article we obtain a power-saving bound on the exceptional set of triples. Specifically, we show that there are $O(X^{33/50})$ integer triples $(a,b,c)\in [1,X]^3$, which satisfy $\mathrm{rad}(abc) < c^{1-ε}$. The proof is based on a combination of bounds for the density of integer points on varieties, coming from the determinant method, Thue equations, geometry of numbers, and Fourier analysis.

Theorems & Definitions (28)

• Proposition 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• proof : Proof of Proposition \ref{['prop:diophantine']}
• Proposition 3.1: Fourier analysis bound
• proof
• Proposition 3.2: Geometry of numbers bound
• proof