On the exceptional set in the $abc$ conjecture

The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and rad(abc) < c1-ε for any $ε> 0$. Using the optimized methods in a recent work of Browning, Lichtman and Teräväinen, we showed that the number of those triples with c X is O(X56/85+ε) for any ε > 0, where 56/85 ≈ 0.658824. This constitutes an improvement of the previous bound O(X33/50).