zbMATH — the first resource for mathematics

On simultaneous rational approximation to a \(p\)-adic number and its integral powers. (English) Zbl 1264.11058
Let \(p\) denote a prime number and \(|\,\cdot\,|_p\) the usual \(p\)-adic value normalized by \(|\,p\,|_p=p^{-1}\). Let \(n\geq 1\) be an integer and \(\xi\) a \(p\)-adic integer.
Mahler introduced the exponent of Diophantine approximation \(w_n\). Let us denote \(\underline{w_n(\xi)}\) the supremum of the real numbers \(w\) such that, for arbitrary large real numbers \(X\), the inequalities \(0<|x_n\xi^n+\dots+x_1\xi+x_0|_p\leq X^{-w-1}, \;\;\max_{0\leq m\leq n}|x_m|\leq X\) have a solution in integers \(x_0,\dots,x_n\).
Another exponent of Diophantine approximation measuring the quality of the simultaneous rational approximation to a number and its first \(n\) integral powers has been introduced recently in [Y. Bugeaud and M. Laurent, Ann. Inst. Fourier 55, No. 3, 773–804 (2005; Zbl 1155.11333)] for the real case: Let us denote \(\underline{\lambda_n(\xi)}\) the supremum of the real numbers \(\lambda\) such that, for arbitrarily large real numbers \(X\), the inequalities \(0<|x_0|\leq X,\;\max_{1\leq m\leq n} |x_0\xi^m-x_m|_p\leq X^{-\lambda-1}\) have a solution in integers \(x_0,\dots,x_n\). It is known that \(\lambda_n(\xi)\geq 1/n\) from \(p\)-adic version of the Dirichlet theorem, \(\lambda_n(\xi) =\max\{1/n,1/(d-1)\}\) for every positive integer \(n\) and every \(p\)-adic algebraic number \(\xi\) of degree \(d\), and \(\lambda_n(\xi)=1/n\) for almost every \(p\)-adic integer \(\xi\).
The spectrum of a function is the set of values taken by this function on the set of transcendental \(p\)-adic numbers.
The authors address the following question: Let \(n\geq 1\) be an integer. Is the spectrum of the function \(\lambda_n\) equal to \([1/n,\infty]\)?
At first, the authors give the following \(p\)-adic analogue of a result in [Y. Bugeaud, Ann. Inst. Fourier 60, No. 6, 2165–2182 (2010; Zbl 1229.11100)]: let \(n\geq 1\) be an integer. For any real number \(w\geq 2n-1\), there exists uncountably many \(p\)-adic integers \(\xi\) such that \(w_1(\xi)=\dots =w_n(\xi)=w\).
Proceeding as in [N. Budarina, D. Dickinson and J. Levesley, Mathematika 56, No. 1, 77–85 (2010; Zbl 1279.11076)], using K. Mahler’s transference principle [Čas. Mat. Fys. 68, 85–92 (1939; Zbl 0021.10402 and JFM 65.0177.01)], they prove : Let \(n\geq 1\) be an integer and \(\lambda\geq 1\) be a real number. There are uncountably many \(p\)-adic integers \(\xi\), which can be constructed explicitly, such that \(\lambda_n(\xi)=\lambda\). In particular, the spectrum of \(\lambda_n\) contains the interval \([1,\infty]\).
A result in [V. I. Bernik and M. M. Dodson, Metric Diophantine approximation on manifolds. Cambridge: Cambridge University Press (1999; Zbl 0933.11040)] on Hausdorff dimension of the set of \(p\)-adic numbers \(\xi\) with a prescribed value for \(\lambda_n(\xi)\) is for \(n=1\), \(\dim\{\xi\in \mathbb Q_p : \lambda_1(\xi)=\lambda\}=\frac{2}{1+\lambda}\). The authors generalize to: Let \(n\geq 2\) be an integer. Let \(\lambda>n-1\) be a real number. Then \(\dim\{\xi\in \mathbb Q_p : \lambda_n(\xi)=\lambda\}=\frac{2}{n(1+\lambda)}\).

11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations
Full Text: DOI
[1] DOI: 10.1112/S0025579309000382 · Zbl 1279.11076 · doi:10.1112/S0025579309000382
[2] Bugeaud, Cambridge Tracts in Mathematics 160 (2004)
[3] DOI: 10.4007/annals.2007.166.367 · Zbl 1137.11048 · doi:10.4007/annals.2007.166.367
[4] Beresnevich, Memoirs of the American Mathematical Society 846 (2006)
[5] DOI: 10.1007/s00222-006-0509-9 · Zbl 1185.11047 · doi:10.1007/s00222-006-0509-9
[6] Sprindžuk, Translations of Mathematical Monographs 25 (1969)
[7] Robert, Graduate Texts in Mathematics 198 (2000)
[8] DOI: 10.1007/BF01092785 · doi:10.1007/BF01092785
[9] Mahler, Časopis Pst. Mat. Fys. 68 pp 85– (1939)
[10] Mahler, Mathematica (Zutphen) pp 177– (1935)
[11] DOI: 10.1515/crll.1968.232.122 · Zbl 0174.08503 · doi:10.1515/crll.1968.232.122
[12] Mahler, Nieuw Arch. Wisk. 18 pp 22– (1934)
[13] Bugeaud, Diophantine Geometry 4 pp 101– (2007)
[14] DOI: 10.5802/aif.2114 · Zbl 1155.11333 · doi:10.5802/aif.2114
[15] Bugeaud, Annales Inst. Fourier 60 pp 2165– (2010) · Zbl 1229.11100 · doi:10.5802/aif.2580
[16] Bernik, Cambridge Tracts in Mathematics 137 (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.