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A proof of the matrix version of Baker's conjecture in Diophantine approximation

Published online by Cambridge University Press:  07 May 2018

TUSHAR DAS  and
DAVID SIMMONS
TUSHAR DAS
Affiliation:
University of Wisconsin–La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI 54601, U.S.A. e-mail: [email protected]
DAVID SIMMONS
Affiliation:
University of York, Department of Mathematics, Heslington, York YO10 5DD. e-mail: [email protected]
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Abstract

We prove that the matrix analogue of the Veronese curve is strongly extremal in the sense of Diophantine approximation, thereby resolving a question posed by Beresnevich, Kleinbock and Margulis (2015) in the affirmative.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 159 - 169
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Aka, M., Breuillard, E., Rosenzweig, L. and de Saxcé, N. On metric Diophantine approximation in matrices and Lie groups. C. R. Math. Acad. Sci. Paris 353 (2015), no. 3, 185–189.Google Scholar
[2] Baker, A. Transcendental Number Theory (Cambridge University Press, London-New York, 1975).Google Scholar
[3] Basu, S., Pollack, R. and Roy, M–F. Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10 (Springer, Berlin, 2006).Google Scholar
[4] Beresnevich, V., Kleinbock, D. and Margulis, G. Non-planarity and metric Diophantine approximation for systems of linear forms. J. Théor. Nombres Bordeaux 27 (2015), no. 1, 1–31.Google Scholar
[5] Bernik, V. and Dodson, M. Metric Diophantine approximation on manifolds. Cambridge Tracts in Mathematics, vol. 137 (Cambridge University Press, Cambridge, 1999).Google Scholar
[6] Das, T., Fishman, L., Simmons, D. and Urbański, M. Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math. (N.S.) (2017), in press.Google Scholar
[7] Kleinbock, D. An ‘almost all versus no’ dichotomy in homogeneous dynamics and Diophantine approximation. Geom. Dedicata 149 (2010), 205–218.Google Scholar
[8] Kleinbock, D., Lindenstrauss, E. and Weiss, B. On fractal measures and Diophantine approximation. Selecta Math. (N.S.) 10 (2004), 479–523.Google Scholar
[9] Kleinbock, D. and Margulis, G. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), no. 1, 339–360.Google Scholar
[10] Kleinbock, D. and Margulis, G. Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999), no. 3, 451–494.Google Scholar
[11] Kleinbock, D., Margulis, G. and Wang, J. Metric Diophantine approximation for systems of linear forms via dynamics. Int. J. Number Theory 6 (2010), no. 5, 1139–1168.Google Scholar
[12] Mahler, K. Über das Maß der Menge aller S-Zahlen. Math. Ann. 106 (1932), no. 1, 131–139 (German).Google Scholar
[13] Sprindžuk, V. G. A proof of Mahler's conjecture on the measure of the set of S-numbers. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 379–436 (Russian).Google Scholar
[14] Sprindžuk, V. G. Achievements and problems of the theory of Diophantine approximations. Uspekhi Mat. Nauk 35 (1980), no. 4(214), 3–68, 248 (Russian).Google Scholar
[15] Stratmann, B. and Urbański, M. Diophantine extremality of the Patterson measure. Math. Proc. Cambridge Philos. Soc. 140 (2006), 297–304.Google Scholar
[16] Urbański, M. Diophantine approximation and self-conformal measures. J. Number Theory 110 (2005), 219–235.Google Scholar