Rational approximations on toric varieties

Zhìzhōng Huáng

Motivated by the local behaviour of rational points on varieties over number fields, McKinnon and M. Roth introduced the notion “approximation constant” attached to every rational point. Roughly speaking, for a given rational point \(Q\) and a place \(v\), it measures how quickly the height must grow for any sequence of rational points approaching \(Q\) with respect to a \(v\)-adic distance. A conjecture of McKinnon predicts that rational curves should be the unique medias on which this constant is achieved. Building on Salberger’s universal torsor method and Batyrev’s toric “Bend-and-Break”, we verify McKinnon’s conjecture for certain split toric varieties and we give the precise locally accumulating subvariety.