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Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 333–360 | Cite as

Threshold functions and Poisson convergence for systems of equations in random sets

  • Juanjo Rué
  • Christoph Spiegel
  • Ana Zumalacárregui
Article
  • 65 Downloads

Abstract

We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets, \(B_{h}[g]\)-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “\(\mathcal {A}\) contains a non-trivial solution of \(M\cdot \mathbf{x }=\mathbf 0 \)” where \(\mathcal {A}\) is a random set and each of its elements is chosen independently with the same probability from the interval of integers \(\{1,\dots ,n\}\). Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.

Notes

Acknowledgements

The authors thank Christian Elsholtz for providing many references concerning linear systems of equations. We also thank Katy Beeler, Arnau Padrol and Lluis Vena for valuable input and fruitful discussions as well as Javier Cilleruelo for a detailed reading of the manuscript and support. Furthermore, we are thankful to the anonymous referees for detailed feedback and constructive suggestions. We would like to thank the Combinatorics and Graph Theory group at the Freie Universität Berlin where part of this research took place for working conditions and hospitality.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Juanjo Rué
    • 1
  • Christoph Spiegel
    • 1
  • Ana Zumalacárregui
    • 2
  1. 1.Department of MathematicsUniversitat Politècnica de Catalunya and Barcelona Graduate School of MathematicsBarcelonaSpain
  2. 2.Department of Pure MathematicsUniversity of New South WalesSydneyAustralia

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