Phase Transitions for Sparse Random Sets Under Linear Forms

TL;DR

This paper investigates phase transitions in the size and multiplicities of linear image sets of sparse random subsets of integers, revealing two distinct threshold scales with precise asymptotics and settling a longstanding conjecture.

Contribution

It identifies and characterizes two separate threshold scales for global size and local multiplicities of linear image sets, providing sharp asymptotics and unifying classical models.

Findings

Global transition at p(N) ~ N^{-(h-1)/h} for size of L(A)

Local transition at p(N) ~ N^{-(h-2)/(h-1)} for representation counts

Poisson behavior of representations below the local threshold

Abstract

Let $A \subseteq \{0,1,\dots,N\}$ be a random set in which each element is included independently with probability $p=p(N)$. Fix an integer $h \geq 2$ and a linear form $$L(x_1,\dots,x_h) := u_1x_1 + \cdots + u_hx_h.$$ We study the random image set \begin{align*} L(A) = \left\{ L(a_1,\dots,a_h) : a_i \in A \right\}, \end{align*} inside the feasible interval of values of $L$ on $\{0,1,\dots,N\}^h$, as well as the associated representation counts. Our results exhibit two distinct threshold scales. First, there is a \emph{global} transition at $p(N) \asymp N^{-(h-1)/h}$ governing the size of $L(A)$: below this scale collisions are rare and $L(A)$ is sparse, while above it $L(A)$ contains nearly all feasible values. We give sharp asymptotics for the size of $L(A)$ in all regimes, including inside the critical window. Second, there is a \emph{local} transition at $p(N)\asymp N^{-(h-2)/(h-1)}$ governing multiplicities: for typical values in the bulk, the number of essentially distinct representations is asymptotically Poisson below this scale, and Poisson behavior fails above it. For $h \geq 3$ these scales are separated, yielding a regime in which $L(A)$ is already globally close to full while local multiplicities remain approximately Poisson. Our framework subsumes the classical sumset and difference-set models, as well as generalized sumsets of the form $sA-dA$, as special cases. Notably, after correcting its formulation, our global theorem settles the 2009 threshold conjecture of Hegarty-Miller on the behavior of these random images.