Multiple Legendre polynomials in Diophantine approximation

TL;DR

This paper introduces a new class of multiple Legendre polynomials satisfying an Apéry-like recurrence, providing improved bounds on approximation measures of logarithms of rationals by algebraic numbers, and refining the nonquadraticity exponent of log 2.

Contribution

It constructs novel multiple Legendre polynomials with specific recurrence properties and derives tighter bounds on approximation measures, advancing Diophantine approximation techniques.

Findings

Bound on the nonquadraticity exponent of log 2 is improved to 12.841618.

New upper bounds for approximation measures of logarithms of rational numbers.

Construction of multiple Legendre polynomials satisfying an Apéry-like recurrence.

Abstract

We construct a class of multiple Legendre polynomials and prove that they satisfy an Ap\'ery-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree. We prove e.g. that the nonquadraticity exponent of $\log 2$ is bounded from above by $12.841618...$, thus improving upon a recent result of the author. Our construction also yields some other known results.