The Lang–Vojta conjecture for integral points

Let $k$ be a number field and let $X/k$ be a smooth projective variety with canonical sheaf $\mathcal{K}$. Let $D$ be an effective, reduced, normal crossings divisor on $X$, and let $S$ be any finite set of primes in $\mathcal{O}_k$. If

$$\mathcal{K}\otimes\mathcal{O}_X(D)$$

is big, then any set of $(D,S)$-integral points in $X$ is Zariski degenerate, meaning that it is not dense in the Zariski topology.

The Lang–Vojta conjecture. Under these hypotheses, every set of $(D,S)$-integral points in $X$ is Zariski degenerate. These problems are central in Diophantine geometry and, for the most part, they remain open, even when $X=\mathbb{P}^2$ and $D$ is a smooth plane curve.