The metric theory of mixed type linear forms

International Journal of Number Theory

Volumn 9, Issue 1, 2013, Pages 77-90

  Dickinson, Detta ^a   Hussain, Mumtaz ^b  

^a NATIONAL UNIVERSITY OF IRELAND   (Ireland)

^b LA TROBE UNIVERSITY   (Australia)

Author keywords

Diophantine approximation; Hausdorff dimension; Hausdorff measure; Khintchine type theorems; system linear forms

Indexed keywords

EID: 84872703036     PISSN: 17930421     EISSN: None     Source Type: Journal    
DOI: 10.1142/S1793042112501278     Document Type: Article

References (20)

1. V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, 1983), Translated by J. Szücs.
2. V. Beresnevich, H. Dickinson and S. L. Velani, Sets of exact 'logarithmic order' in the theory of Diophantine approximation, Math. Ann. 321(2) (2001) 253-273.
3. -, Measure Theoretic Laws for lim sup Sets, Memoirs of the American Mathe- matical Society, Vol. 179, No. 846 (American Mathematical Society, Providence, RI, 2006).
4. V. Beresnevich and S. L. Velani, Schmidt's theorem, Hausdorff measures and slicing, Int. Math. Res. Notices 2006 (2006) 1-24.
5. -, Ubiquity and a general logarithm law for geodesics, in Dynamical Systems and Diophantine Approximation, Śeminaires et Congr̀es, Vol. 19 (Soc Math. France, Paris, 2009), pp. 21-36.
6. -, Classical metric Diophantine approximation revisited: The Khintchine- Groshev theorem, Int. Math. Res. Notices 2010 (2010) 69-86.
7. V. I. Bernik and M. M. Dodson, Metric Diophantine Approximation on Manifolds, Cambridge Tracts in Mathematics, Vol. 137 (Cambridge University Press, 1999).
8. J. W. S. Cassels, An Introduction to Diophantine Approximation (Cambridge University Press, 1957).
9. H. Dickinson, The Hausdorff dimension of systems of simultaneously small linear forms, Mathematika 40 (1993) 367-374.
10. , The Hausdorff dimension of sets arising in metric Diophantine approxima- tion, Acta Arith. 68(2) (1994) 133-140.
11. H. Dickinson and S. Velani, Hausdorff measure and linear form, J. Reine Angew. Math. 490 (1997) 1-36.
12. M. M. Dodson, B. P. Rynne and J. A. G. Vickers, Diophantine approximation and a lower bound for Hausdorff dimension, Mathematika 37 (1990) 59-73.
13. , The Hausdorff dimension of exceptional sets associated with normal forms, J. London Math. Soc (2) 49 (1994) 614-624.
14. K. J. Falconer, Fractal Geometry \- Mathematical Foundations and Applications (Wiley, 1990).
15. A. V. Groshev, Une theoreme sur les systemes des formes lineaires, Doklady Akad. Nauk SSSR 19 (1938) 151-152.
16. G. Harman, Metric Number Theory, London Mathematical Society Monographs, Vol. 18 (Oxford University Press, 1998).
17. M. Hussain and J. Levesley, The metrical theory of simultaneously small linear forms, to appear in Funct. Approx. Comment. Math.
18. V. Jarník, Uber die simultanen Diophantischen appoximationen, Math. Z. 33 (1931) 505-543.
19. A. Khinthcine, Einige S̈atze über Kettenbrüche, mit anwendungen auf die theorie der Diophantischen approximationen, Math. Ann. 92 (1924) 115-125.
20. V. Sprindzuk, Metric Theory of Diophantine Approximation (Wiley, 1979).