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

Threshold functions and Poisson convergence for systems of equations in random sets 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, $$B_{h}[g]$$ B h [ g ] -sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “ $$\mathcal {A}$$ A contains a non-trivial solution of $$M\cdot \mathbf{x }=\mathbf 0 $$ M · x = 0 ” where $$\mathcal {A}$$ A is a random set and each of its elements is chosen independently with the same probability from the interval of integers $$\{1,\dots ,n\}$$ { 1 , ⋯ , 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

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

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1891-2
Publisher site
See Article on Publisher Site

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, $$B_{h}[g]$$ B h [ g ] -sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “ $$\mathcal {A}$$ A contains a non-trivial solution of $$M\cdot \mathbf{x }=\mathbf 0 $$ M · x = 0 ” where $$\mathcal {A}$$ A is a random set and each of its elements is chosen independently with the same probability from the interval of integers $$\{1,\dots ,n\}$$ { 1 , ⋯ , 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.

Journal

Mathematische ZeitschriftSpringer Journals

Published: May 10, 2017

References

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