Note on rational approximations of the exponential function at 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...

A Note on the Rational Points

Let C be the image of a canonical embedding φ of the Atkin-Lehner quotient X + 0 (N) associated to the Fricke involution w N. Suppose φ is defined over the rationals. In this note we give some collinearity relations among rational points of C, for each X + 0 (N) of genus 3 and the first X + 0 (N) of genus 4, for N prime.

A Note on the Rational Points of X+0(N)

Let C be the image of a canonical embedding φ of the Atkin-Lehner quotient X 0 (N) associated to the Fricke involution wN . In this note we exhibit some relations among the rational points of C. For each g = 3 (resp. the first g = 4) curve C we found that there are one or more lines (resp. planes) in P whose intersection with C consists entirely of rational Heegner points or the cusp point, whe...

Rational Interpolation of the Exponential Function

Let m, n be nonnegative integers and a<m+n) be a set of m + n + 1 real in~lation points (not necessarily distinct). Let Rm.n = P m.n / Qm.n be the unique rational function with degPm.n ::; m, deg Qm,n :$n, that in~lates e'" in the points of a<nr+n). Ifm = mv, n = nv with mv + nv --+ 00, and mv /nv --+ A as v --+ 00, and the sets a<nr+n) are uniformly bounded, we show that Pm.n(z) --+ ~/(l+)'), ...

Distribution of Rational Points: A Survey