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arXiv · 2512.12890

Multiple Legendre polynomials in Diophantine approximation

Raffaele Marcovecchio
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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.

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Raffaele Marcovecchio. 2025-12-15. Multiple Legendre polynomials in Diophantine approximation. https://doi.org/10.1142/s1793042114500584

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