Phase Transitions for Sparse Random Sets Under Linear Forms

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.