The abc conjecture almost always

The celebrated abc conjecture asserts that every solution to the equation a+b=c in triples of coprime integers (a,b,c) must satisfy rad(abc) >= c^{1-\epsilon}, with finitely many exceptions. We prove a power-saving bound on the exceptional set of such triples. Namely, there are at most O(X^{33/50}) many triples of coprime integers in a cube (a,b,c)\in [1,X]^3 satisfying a+b=c and rad(abc) < c^{1-\epsilon}. The proof is based on a combination of bounds for the density of integer points on varieties. Joint work with Tim Browning and Joni Teravainen.