A conjecture on rational approximations to rational points

A Conjecture on Rational Approximations to Rational Points

In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta’s Main Conj...

Distribution of Rational Points: A Survey

Rational Approximations to Rational Models of Categorization

Can statistical machine learning theories and algorithms help explain human learning? Broadly speaking, machine learning studies the fundamental laws that govern all learning processes, including both artificial systems (e.g., computers) and natural systems (e.g., humans). It has long been understood that theories and algorithms from machine learning are relevant to understanding aspects of hum...

Rational Approximations to n

Using an IBM 1130 computer, we have generated the first 20,000 partial quotients in the ordinary continued-fraction representation of x.

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...