Upper bounds for the uniform simultaneous Diophantine exponents

Abstract

We give several upper bounds for the uniform simultaneous Diophantine exponent λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ of a transcendental number ξ∈double-struckR$\xi \in \mathbb {R}$. The most important one relates λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ and the ordinary simultaneous exponent ωk(ξ)$\omega _k(\xi )$ in the case when k is substantially smaller than n. In particular, in the generic case ωk(ξ)=k$\omega _k(\xi )=k$ with a properly chosen k, the upper bound for λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ becomes as small as 32n+O(n−2)$\frac{3}{2n} + O(n^{-2})$ which is substantially better than the best currently known unconditional bound of 2n+O(n−2)$\frac{2}{n} + O(n^{-2})$. We also improve an unconditional upper bound on λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ for even values of n.

On geometry of numbers and uniform rational approximation to the Veronese curve

“…For even n, the author improved the bound from [5] in [16,17]. Then in a very recent paper Badziahin [1] improved on the previous results for n ≥ 4, which in turn has been refined by Poels and Roy [11] to constitute the currently best known bounds for λ n (ξ). In private communication, D. Roy pointed out to me that he recently obtained a very small improvement on his result [13] for n = 3, in a paper in preparation.…”

“…The polynomial P (1) minimizes L P over all relevant choices of P , hence L 1 (q) = min L P (q) = min max log H P − q 2n − 2 , log |P (ξ)| + q with minimum taken over all non-zero integer polynomials P of degree at most 2n − 2.…”

“…It is well-known that w 1 (ξ) = 1 for all irrational real numbers ξ, see Khintchine [6], so we may restrict to n ≥ 2. As indicated above, upper bounds for w n (ξ) have first been studied by Davenport and Schmidt [5], whose result shows in our notation that (1) w n (ξ) ≤ 2n − 1, n ≥ 2, holds for any real ξ. For n = 2, they proved a stronger bound of the form…”