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Vojta's Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces.