Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T01:24:01.787Z Has data issue: false hasContentIssue false
  • English
  • Français

RESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory Exponential sums and character sums

Published online by Cambridge University Press:  26 July 2016

Stephan Baier  and
Anish Ghosh
Stephan Baier
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, 400005, India email [email protected]
Anish Ghosh
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, 400005, India email [email protected]
Get access

Abstract

We study the problem of Diophantine approximation on lines in $\mathbb{R}^{d}$ under certain primality restrictions.

MSC classification

Primary: 11J83: Metric theory 11K60: Diophantine approximation 11L07: Estimates on exponential sums
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 34 - 52
Copyright
Copyright © University College London 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beresnevich, V., Bernik, V. I., Dickinson, D. and Dodson, M. M., On linear manifolds for which the Khinchin approximation theorem holds. Vesti Nats Akad. Navuk Belaruis Ser. Fiz.-mat. Navuk 2 2000, 14–17.Google Scholar
Baier, S. and Ghosh, A., Diophantine approximation on lines with prime constraints. Q. J. Math. 66(1) 2015, 1–12.Google Scholar
Brüdern, J., Einführung in die analytische Zahlentheorie, Springer (Berlin and New York, 1991).Google Scholar
Cassels, J. W. S., An Introduction to Diophantine Approximation, Cambridge University Press (1957).Google Scholar
Ghosh, A., A Khintchine-type theorem for hyperplanes. J. Lond. Math. Soc. (2) 72(2) 2005, 293–304.Google Scholar
Ghosh, A., Diophantine exponents and the Khintchine Groshev theorem. Monatsh. Math. 163(3) 2011, 281–299.Google Scholar
Graham, S. W. and Kolesnik, G., Van der Corput’s Method of Exponential Sums (London Mathematical Society Lecture Notes Series 126 ), Cambridge University Press (Cambridge, 1991).CrossRefGoogle Scholar
Halberstam, H. and Richert, H.-R., Sieve Methods, Academic Press (New York/London, 1974).Google Scholar
Harman, G., Metric diophantine approximation with two restricted variables III. Two prime numbers. J. Number Theory 29 1988, 364–375.CrossRefGoogle Scholar
Harman, G., Small fractional parts of additive forms. Philos. Trans. R. Soc. Lond. A 355 1993, 327–338.Google Scholar
Harman, G., Metric Number Theory (London Mathematical Society Monographs New Series, 18 ), Oxford University Press (Oxford, 1998).CrossRefGoogle Scholar
Harman, G. and Jones, H., Metrical theorems on restricted diophantine approximations to points on a curve. J. Number Theory 97(1) 2002, 45–57.Google Scholar
Heath-Brown, D. R. and Jia, C., The distribution of 𝛼p modulo one. Proc. Lond. Math. Soc. (3) 84(1) 2002, 79–104.CrossRefGoogle Scholar
Jones, H., Khintchins theorem in k dimensions with prime numerator and denominator. Acta Arith. 99 2001, 205–225.CrossRefGoogle Scholar
Kleinbock, D., Extremal subspaces and their submanifolds. Geom. Funct. Anal. 13(2) 2003, 437–466.Google Scholar
Matomäki, K., The distribution of 𝛼p modulo one. Math. Proc. Cambridge Philos. Soc. 147 2009, 267–283.Google Scholar
Ramachandra, K., Two remarks in prime number theory. Bull. Soc. Math. France 105(4) 1977, 433–437.Google Scholar
Srinivasan, S., A note on |𝛼p - q|. Acta Arith. 41 1982, 15–18.Google Scholar
Vaaler, J. D., Some extremal problems in Fourier analysis. Bull. Amer. Math. Soc. 12 1985, 183–216.Google Scholar
Vaughan, R. C., Sommes trigonométriques sur les nombres premiers. C. R. Acad. Sci. Paris Sér. A 285 1977, 981–983.Google Scholar