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Phase Transitions for Sparse Random Sets Under Linear Forms

Ryan Jeong, Steven J. Miller

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.

Phase Transitions for Sparse Random Sets Under Linear Forms

Abstract

Let be a random set in which each element is included independently with probability . Fix an integer and a linear form 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 on , as well as the associated representation counts. Our results exhibit two distinct threshold scales. First, there is a \emph{global} transition at governing the size of : below this scale collisions are rare and is sparse, while above it contains nearly all feasible values. We give sharp asymptotics for the size of in all regimes, including inside the critical window. Second, there is a \emph{local} transition at 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 these scales are separated, yielding a regime in which 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 , 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.
Paper Structure (43 sections, 24 theorems, 231 equations, 1 figure)

This paper contains 43 sections, 24 theorems, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Notation and Conventions
  4. Linear forms, images, and complements
  5. Injective solutions modulo redundancies
  6. Random model and probabilistic notation
  7. The Irwin-Hall density and the critical profile
  8. Global assumptions and notation
  9. Main Results
  10. Global Phase Transition
  11. Local Poisson Convergence
  12. Auxiliary Remarks
  13. Proof Outline
  14. Organization
  15. Preliminaries
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $p : \mathbb{N} \to (0,1)$ be a function satisfying eq:p_assumptions. Fix an integer $h \geq 2$ and a linear form $L: \mathbb{Z}^h \to \mathbb{Z}$ with coefficients $u_1, \dots, u_h \in \mathbb{Z}_{\neq 0}$ such that $\gcd(u_1, \dots, u_h) = 1$. Let $A \subseteq I_N$ be a random subset where eac

Figures (1)

  • Figure 1: Schematic phase diagram (shown for $h = 4$) illustrating \ref{['thm:Z_linear_forms', 'thm:poisson_convergence']}. The solid curve $p(N) = N^{-(h-1)/h}$ represents the global threshold for the coverage of $L(A)$ from \ref{['thm:Z_linear_forms']}, while the dashed curve $p(N) = N^{-(h-2)/(h-1)}$ represents the local threshold for Poisson approximation of bulk representation counts from \ref{['thm:poisson_convergence']}. Below the solid curve, $L(A)$ is sparse within its feasible interval and is "essentially Sidon." The shaded band indicates the intermediate regime in which the random image $L(A)$ occupies nearly all of its feasible range, but bulk representation counts remain well-approximated by a Poisson law. The hatched region indicates densities above the local threshold where dependence from overlaps is no longer negligible and Poisson approximation fails. The horizontal axis is logarithmic in $N$ and the vertical axis is linear in $p$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4: barbour1992poisson, Equation (1.1)
  • Theorem 2.5: arratia1989two, Theorem 1
  • Theorem 2.6: alon2016probabilistic
  • Theorem 2.7: arratia1989two, Theorems 1 and 2
  • ...and 31 more