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

TL;DR

This paper develops a unified framework to analyze threshold functions for the existence of solutions to linear systems in random sets, extending previous definitions and studying Poisson convergence of solution counts.

Contribution

It introduces a comprehensive approach to threshold functions for various combinatorial structures and characterizes the distribution of solutions at the threshold, including Poisson convergence.

Findings

Existence of a threshold function for solutions in random sets.

Distribution of non-trivial solutions converges to a Poisson distribution.

Threshold behavior depends on convex polytope volumes and structural symmetries.

Abstract

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets 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 $M\cdot\textbf{x}=\textbf{0}$", 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 combinatorial structure, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the behaviour of 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.