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We generalize the study of the sum and difference sets of a subset of$\mathbb{N}$ drawn from a binomial model to the following setting. Given $A\subseteq \{0, 1, \dots, N\}$, an integer $h \geq 2$, and a linear form $L:\mathbb{Z}^h \to \mathbb{Z}$ given by $$L(x_1, \dots, x_h) = u_1x_1 + \cdots +u_hx_h, \quad u_i \in \mathbb{Z}_{\neq 0} \text{ for all } i \in [h],$$ westudy the size of $$L(A) = \left\{u_1a_1 + \cdots + u_ha_h : a_i \in A\right\}$$ and its complement $L(A)^c$ when each element of $\{0, 1, \dots,N\}$ is independently included in $A$ with probability $p(N)$. We identify twophase transition phenomena. The first ``global" phase transition concerns therelative sizes of $L(A)$ and $L(A)^c$, with $p(N) = N^{-\frac{h-1}{h}}$ as thethreshold. Asymptotically almost surely, it holds below the threshold thatalmost all sums generated in $L(A)$ are distinct and almost all possible sumsare in $L(A)^c$, and above the threshold that almost all possible sums are in$L(A)$. Our asymptotic formulae substantially extend work of Hegarty and Millerand completely settle, with appropriate corrections made to its statement,their conjecture from 2009. The second ``local" phase transition concerns theasymptotic behavior of the number of distinct realizations in $L(A)$ of a givenvalue, with $p(N) = N^{-\frac{h-2}{h-1}}$ as the threshold. Specifically, itidentifies (in a sharp sense) when the number of such realizations obeys aPoisson limit. Our main tools are recent results concerning the asymptoticenumeration of partitions and weak compositions, classical theorems on Poissonapproximation, and the martingale machinery of Kim and Vu.

Phase Transitions for Binomial Sets Under Linear Forms