2021/07/23 by Dzmitry Badziahin, Badziahin, Dmitry
Mathematics · #11J13 #11J82 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2107.11134
openalex publication_date 2021/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give several upper bounds for the uniform simultaneous Diophantine exponent \widehatλn(ξ) of a transcendental number ξ∈ℝ. The most important one relates \widehatλn(ξ) and the ordinary simultaneous exponent ωk(ξ) in the case when k is substantially smaller than n. In particular, in the generic case ωk(ξ)=k with a properly chosen k, the upper bound for \widehatλn(ξ) becomes as small as (3)/(2n) + O(n-2) which is substantially better than the best currently known unconditional bound of (2)/(n) + O(n-2). We also improve an unconditional upper bound on \widehatλn(ξ) for even values of n.