Approximating rational points on toric varieties

Given a smooth projective variety X over a number field k and P\in X(k), the first author conjectured that in a precise sense, any sequence that approximates P sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if X is a \mathbb{Q}-factorial terminal split toric variety of arbitrary dimension, then P is better approximated by points on a rational curve than by any Zariski dense sequence.