Simultaneous rational approximation to successive powers of a real number

Abstract

We develop new tools leading, for each integer n ≥ 4 n\ge 4 , to a significantly improved upper bound for the uniform exponent of rational approximation λ ^ n ( ξ ) \widehat {\lambda }_n(\xi ) to successive powers 1 , ξ , … , ξ n 1,\xi ,\dots ,\xi ^n of a given real transcendental number ξ \xi . As an application, we obtain a refined lower bound for the exponent of approximation to ξ \xi by algebraic integers of degree at most n + 1 n+1 . The new lower bound is n / 2 + a n + 4 / 3 n/2+a\sqrt {n}+4/3 with a = ( 1 − log ⁡ ( 2 ) ) / 2 ≃ 0.153 a=(1-\log (2))/2\simeq 0.153 , instead of the current n / 2 + O ( 1 ) n/2+\mathcal {O}(1) .

Upper bounds for the uniform simultaneous Diophantine exponents

“…In general, one can check that $$t_n < \tau _{2n}$$ for all $$n\geqslant 2$$. Remark After this paper was initially submitted for publication, Poëls and Roy [10] announced stronger upper bounds for $$\widehat{\lambda }_n(\xi )$$ than in this paper. In particular, they got $$\widehat{\lambda }_4(\xi )\leqslant 0.3370\ldots$$ and $$\widehat{\lambda }_6(\xi )\leqslant 0.2444\ldots$$ The Diophantine exponent $$\widehat{\lambda }_3$$ was studied in more detail by Roy [12] where he got a better upper bound $$\widehat{\lambda }_3(\xi ) \leqslant \lambda _0\approx 0.4245$$ than for general n .…”

“…\end{equation*}We can see that in Example 2, Theorem 4 gives better upper bounds for $$\widehat{\lambda }_n(\xi )$$ for all values of $$n\geqslant 7$$. Remark In light of [10] which appeared after this paper was written, some of the upper bounds from both examples are now superseded. However, the upper bound on $$\widehat{\lambda }_4(\xi )$$ from Example 1 and on $$\widehat{\lambda }_{2m+3}(\xi )$$ from Example 2 are still best currently known.…”

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

“…All error terms below will be understood as k → ∞. Then, upon assuming (10), by (11) and a well-known argument (see for example [18, Lemma 1]) we have (19) 1…”

“…Remark 4. If (10) holds, then we may use (11) to eliminate w n (ξ) and express the right hand sides in terms of n, w n (ξ), τ k , τ ℓ only. For example (24) becomes log…”

“…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.…”