Interlacing of alternation points in rational approximation

Distribution of Rational Points: A Survey

On the Abc Conjecture and Diophantine Approximation by Rational Points

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the “abc conjecture” of Masser and Oesterlé. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split fun...

Rational points near manifolds and metric Diophantine approximation

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarńık type theorems for submanifolds of Rn. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this...

Rational Points

is known to have only finitely many triples of positive integer solutions x, y, z for a given n > 2 (Faltings, 1983). In Chapter 11, special situations are described in which more precise information is accessible. For example, if x is in S, then n is bounded by a computable number C5 = Cb(pv ..., p8). From these examples, it should be clear that the book is a mine of information for workers in...

Uniformity of Rational Points

1. Uniformity and correlation 1 1.