# On simultaneous rational approximation to a real number and its integral powers

Yann Bugeaud[1]

• [1] Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg (FRANCE)
• Volume: 60, Issue: 6, page 2165-2182
• ISSN: 0373-0956

top

## Abstract

top
For a positive integer $n$ and a real number $\xi$, let ${\lambda }_{n}\left(\xi \right)$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $||q\xi ||,||q{\xi }^{2}||,...,||q{\xi }^{n}||$ are all less than ${q}^{-\lambda }$. Here, $||·||$ denotes the distance to the nearest integer. We study the set of values taken by the function ${\lambda }_{n}$ and, more generally, we are concerned with the joint spectrum of $\left({\lambda }_{1},...,{\lambda }_{n},...\right)$. We further address several open problems.

## How to cite

top

Bugeaud, Yann. "On simultaneous rational approximation to a real number and its integral powers." Annales de l’institut Fourier 60.6 (2010): 2165-2182. <http://eudml.org/doc/116329>.

@article{Bugeaud2010,
abstract = {For a positive integer $n$ and a real number $\xi$, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi ^2 ||, \ldots , ||q \xi ^n||$ are all less than $q^\{-\lambda \}$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We study the set of values taken by the function $\lambda _n$ and, more generally, we are concerned with the joint spectrum of $(\lambda _1, \ldots , \lambda _n , \ldots )$. We further address several open problems.},
affiliation = {Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg (FRANCE)},
author = {Bugeaud, Yann},
journal = {Annales de l’institut Fourier},
keywords = {Simultaneous approximation; exponent of approximation; simultaneous approximation},
language = {eng},
number = {6},
pages = {2165-2182},
publisher = {Association des Annales de l’institut Fourier},
title = {On simultaneous rational approximation to a real number and its integral powers},
url = {http://eudml.org/doc/116329},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Bugeaud, Yann
TI - On simultaneous rational approximation to a real number and its integral powers
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 6
SP - 2165
EP - 2182
AB - For a positive integer $n$ and a real number $\xi$, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi ^2 ||, \ldots , ||q \xi ^n||$ are all less than $q^{-\lambda }$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We study the set of values taken by the function $\lambda _n$ and, more generally, we are concerned with the joint spectrum of $(\lambda _1, \ldots , \lambda _n , \ldots )$. We further address several open problems.
LA - eng
KW - Simultaneous approximation; exponent of approximation; simultaneous approximation
UR - http://eudml.org/doc/116329
ER -

## References

top
1. B. Adamczewski, Y. Bugeaud, Palindromic continued fractions, Ann. Inst. Fourier (Grenoble) 57 (2007), 1557-1574 Zbl1126.11036MR2364142
2. V. Beresnevich, Rational points near manifolds and metric Diophantine approximation Zbl1264.11063
3. V. Beresnevich, D. Dickinson, S. L. Velani, Diophantine approximation on planer curves and the distribution of rational points, Ann. of Math. 166 (2007), 367-426 Zbl1137.11048MR2373145
4. V. I. Bernik, Application of the Hausdorff dimension in the theory of Diophantine approximations, Acta Arith. 42 (1983), 219-253 Zbl0482.10049MR729734
5. N. Budarina, D. Dickinson, J. Levesley, Simultaneous Diophantine approximation on polynomial curves, Mathematika 56 (2010), 77-85 Zbl1279.11076MR2604984
6. Y. Bugeaud, Approximation by algebraic numbers, (2004), Cambridge University Press Zbl1055.11002MR2136100
7. Y. Bugeaud, Diophantine approximation and Cantor sets, Math. Ann. 341 (2008), 677-684 Zbl1163.11056MR2399165
8. Y. Bugeaud, Multiplicative Diophantine approximation, Dynamical systems and Diophantine Approximation (to appear) Zbl1266.11083
9. Y. Bugeaud, M. Laurent, On transfer inequalities in Diophantine approximation, II Zbl1234.11086
10. Y. Bugeaud, M. Laurent, Exponents of Diophantine Approximation and Sturmian Continued Fractions, Ann. Inst. Fourier (Grenoble) 55 (2005), 773-804 Zbl1155.11333MR2149403
11. Y. Bugeaud, M. Laurent, Exponents of Diophantine approximation, Diophantine Geometry Proceedings 4 (2007), 101-121, Scuola Normale Superiore Pisa, Ser. CRM Zbl1229.11098MR2349650
12. R. Güting, Zur Berechnung der Mahlerschen Funktionen ${w}_{n}$, J. reine angew. Math. 232 (1968), 122-135 Zbl0174.08503MR233776
13. V. Jarník, Über die simultanen Diophantische Approximationen, Math. Z. 33 (1931), 505-543 Zbl57.1370.01MR1545226
14. V. Jarník, Über einen Satz von A. Khintchine II, Acta Arith. 2 (1936), 1-22 Zbl0015.29405
15. D. Kleinbock, E. Lindenstrauss, B. Weiss, On fractal measures and Diophantine approximation, Selecta Math. 10 (2004), 479-523 Zbl1130.11039MR2134453
16. S. Lang, Algebra, 211 (2002), Springer-Verlag, New York Zbl0984.00001MR1878556
17. M. Laurent, On transfer inequalities in Diophantine Approximation, Analytic Number Theory in Honour of Klaus Roth (2009), 306-314, Cambridge University Press Zbl1163.11053MR2508652
18. K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. reine angew. Math. 166 (1932), 118-150 Zbl0003.38805
19. W. M. Schmidt, On heights of algebraic subspaces and diophantine approximations, Annals of Math. 85 (1967), 430-472 Zbl0152.03602MR213301
20. V. G. Sprindžuk, Mahler’s problem in metric number theory, (1967), Nauka i Tehnika, Minsk Zbl0168.29504
21. R. C. Vaughan, S. Velani, Diophantine approximation on planar curves: the convergence theory, Invent. Math. 166 (2006), 103-124 Zbl1185.11047MR2242634
22. E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. reine angew. Math. 206 (1961), 67-77 Zbl0097.03503MR142510

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.