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Metrical Theory for Small Linear Forms and Applications to Interference Alignment

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From Analysis to Visualization (JBCC 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 313))

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Abstract

In this paper, the metric theory of Diophantine approximation associated with mixed type small linear forms is investigated. We prove Khintchine–Groshev type theorems for both the real and complex number systems. The motivation for these metrical results comes from their applications in signal processing. One such application is discussed explicitly.

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Acknowledgements

The authors are listed in alphabetical order. We would like to thank Professor Amir Khandani for many useful discussions and guidance. We thank the anonymous referee and Richard Brent for useful comments which have improved the quality and the presentation of the paper. Mumtaz Hussain would like to thank his late mentor Laureate Professor Jonathan Borwein for many fruitful discussions surrounding topics of this paper in particular and number theory in general during his career at the University of Newcastle (Australia). Jon left a lifelong impression on many of us with his breadth of knowledge, sharpness and innovative ideas on nearly every topic.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, PO Box 199, 3552, Bendigo, Australia

    Mumtaz Hussain & Seyyed Hassan Mahboubi

  2. Department of Computer Engineering, Sharif University of Technology, Tehran, Iran

    Abolfazl Seyed Motahari

Authors
  1. Mumtaz Hussain
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  2. Seyyed Hassan Mahboubi
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  3. Abolfazl Seyed Motahari
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Corresponding author

Correspondence to Mumtaz Hussain .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    David H. Bailey

  2. Western University, London, ON, Canada

    Naomi Simone Borwein

  3. Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia

    Richard P. Brent

  4. School of IT and Mathematical Sciences, University of South Australia, Mawson Lakes, SA, Australia

    Regina S. Burachik

  5. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Judy-anne Heather Osborn

  6. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Brailey Sims

  7. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Qiji J. Zhu

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Hussain, M., Mahboubi, S.H., Motahari, A.S. (2020). Metrical Theory for Small Linear Forms and Applications to Interference Alignment. In: Bailey, D., et al. From Analysis to Visualization. JBCC 2017. Springer Proceedings in Mathematics & Statistics, vol 313. Springer, Cham. https://doi.org/10.1007/978-3-030-36568-4_24

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