Simultaneous rational approximation to successive powers of a real number

2021/09/29 by Poëls, Anthony, Roy, Damien
#FOS: Mathematics #Number Theory (math.NT) #Primary: 11J13 #Secondary: 11J82

paper · doi:10.48550/arxiv.2109.14141

Abstract

We develop new tools leading, for each integer n≥ 4, to a significantly improved upper bound for the uniform exponent of rational approximation \widehatλ_n(ξ) to successive powers 1,ξ,…,ξⁿ of a given real transcendental number ξ. As an application, we obtain a refined lower bound for the exponent of approximation to ξ by algebraic integers of degree at most n+1. The new lower bound is n/2+a√(n)+4/3 with a=(1-log(2))/2≃ 0.153, instead of the current n/2+O(1).

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