Acta mathematica scientia, Series B >
A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION
Received date: 2013-12-04
Revised date: 2014-01-08
Online published: 2015-01-20
Supported by
This work was supported by NSFC (11171124). †Corresponding author: Yu SUN.
Let Q be an infinite set of positive integers, > 1 be a real number and let (Q) = x 2 R :
x − q q− for infinitely many (p, q) 2 Z × Q. For any given positive integer m, set Q(m) = {n 2 N : (n,m) = 1}. If m is divisible by at least two prime factors, Adiceam [1] showed that W (N) \ W (Q(m)) contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers? In this note, we give an affirmative answer to Adiceam’s question.
Key words： liminf set; Diophantine approximation; Liouville number
LIU Jia,SUN Yu . A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION[J]. Acta mathematica scientia, Series B, 2015 , 35(1) : 189 -194 . DOI: 10.1016/S0252-9602(14)60150-8
[1] Adiceam F. A note on the Hausdorff dimension of some liminf sets appearing in simultaneous Diophantine approximation. Mathematika, 2013, 59: 56–64
[2] Adiceam F. Liminf sets in simultaneous Diophantine approximation. 2013, arXiv: 1308.3806[3] Beresnevich V, Dickinson H, Velani S L. Sets of exact “logarithmic order” in the theory of Diophantine approximation. Math Ann, 2001, 321: 253–273
[4] Besicovitch A S. Sets of fractional dimension (IV): on rational approximation to real numbers. J London Math Soc, 1934, 9: 126–131
[5] Borosh I, Fraenkel A S. A generalization of Jarn´?k’s theorem on Diophantine approximations. Nederl Akad Wet Proc, Ser A, 1972, 75: 192–201
[6] Bugeaud Y. Sets of exact approximation order by rational numbers. Math Ann, 2003, 327: 171–190
[7] Bugeaud Y. Approximation by Albebraic Numbers. Cambridge Tracts in Mathematics 160. Cambridge: Cambridge University Press, 2007
[8] Dickinson H, Velani S L. Hausdorff measure and linear forms. J Reine Angew Math, 2007, 490: 1–36
[9] Erd¨os P, Mahler K. Some arithmetical properties of the convergents of a continued fraction. J London Math Soc, 1939, 14: 12–18
[10] Falconer K J. Fractal Geometry. Mathematical Foundations and Application. John Wiley and Sons, 1990
[11] Harman G. Metric Number Theory. London Mathematical Society Monographs, New Seris, 18, 1998
[12] Hinokuma T, Shiga H. Hausdorff dimension of the sets arising in Diophantine approximation. Kodai Math J, 1996, 19: 365–377
[13] Hu D G, Hu X H. Arbitrarily long arithmetic progressions for continued fractions of Laurent series. Acta Math Sci, 2013, 33B(4): 943–949
[14] Jarn´?k V. Diophantischen approximationen und Hausdorffsches mass. Mat Sb, 1929, 36: 371–381
[15] Jarn´?k V. Uber die simultanen diophantischen Approximationen. Math Z, 1931, 33: 503–543
[16] Khintchine A Ya. Continued Fractions. University of Chicago Press, 1964
[17] Schmidt W M. Diophantine Approximation. Lecture notes in Mathematics 785. Berlin, Heidelberg: Springer-Verlag, 1980
/
| 〈 |
|
〉 |