A simultaneous approximation problem for exponentials and logarithms

2025/05/27 by Veekesh Kumar, Kumar, Veekesh, Riccardo Tosi +1
Mathematics · #Mathematical functions and polynomials #Mathematical Approximation and Integration #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.2505.20957

Abstract

Let α₁,α₂ be non-zero algebraic numbers such that (log α₂)/(logα₁)∉ℚ and let β be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials P(x,y,z) and Q(x,y,z) with integer coefficients are not too small at the point ((logα₂)/(log α₁),α₁^β, α₂^β). We also establish a measure of algebraic independence of those numbers among (logα₂)/(log α₁), α^β₁ and α^β₂ which are algebraically independent.

Related

• On the approximation of the values of exponential function and logarithm by algebraic numbers
• On a Problem of Mahler Concerning the Approximation of Exponentials and Logarithms
• B'
• Linear independence of values of logarithms revisited
• Multiple Legendre polynomials in Diophantine approximation