Given a smooth projective variety X X over a number field k k and P ∈ X ( k ) P\in X(k) , the first author conjectured that in a precise sense, any sequence that approximates P 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 X is a Q \mathbb {Q} -factorial terminal split toric variety of arbitrary dimension, then P P is better approximated by points on a rational curve than by any Zariski dense sequence.