Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 333–360

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

Article

## 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.

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© Springer-Verlag Berlin Heidelberg 2017

## Authors and Affiliations

• Juanjo Rué
• 1
Email author
• 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