Nesterenko's criterion when the small linear forms oscillate

TL;DR

This paper extends Nesterenko's criterion to oscillating small linear forms, providing bounds on the dimension of rationally spanned numbers and measures of simultaneous approximation, with applications to zeta values.

Contribution

It generalizes Nesterenko's criterion to oscillating forms, offering new bounds and approximation measures, and applies this to specific zeta values.

Findings

Established a lower bound for the dimension of rationally spanned numbers.

Provided an upper bound for the irrationality exponent of certain numbers.

Applied the criterion to zeta(5), zeta(7), zeta(9), and zeta(11).

Abstract

In this paper we generalize Nesterenko's criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers, and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, and $\zeta(11)$, using Zudilin's proof that at least one of these numbers is irrational.