Normal limiting distributions for systems of linear equations in random sets

TL;DR

This paper demonstrates that the number of solutions to linear systems within random subsets of integers follows a normal distribution across various regimes, under broad conditions on the system and the subset probability.

Contribution

It establishes a general normal limit law for solutions of linear equations in random sets, extending previous results to more general systems and solution types.

Findings

Normal distribution of solution counts across regimes

Applicability to non-trivial solutions in balanced systems

Use of linear algebra and moments method in proof

Abstract

We consider the binomial random set model $[n]_p$ where each element in $\{1,\dots,n\}$ is chosen independently with probability $p:=p(n)$. We show that for essentially all regimes of $p$ and very general conditions for a matrix $A$ and a column vector $\mathbf{b}$, the count of specific integer solutions to the system of linear equations $A\mathbf{x} = \mathbf{b}$ with the entries of $\mathbf{x}$ in $[n]_p$ follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory.