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Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

Sukrit Chakraborty, Sayan Goswami, Sourav Kanti Patra

TL;DR

The paper asks whether random subsets of the natural numbers retain additive and multiplicative structure typical of Hindman-type theorems. It develops and analyzes the Binomial model $\nats_p$, proving that almost surely, FS-sets and FP-sets of every finite length appear, via disjoint-pattern constructions and the second Borel--Cantelli lemma. A novel link between Hindman’s partition theorem and the central limit theorem is established, yielding two limiting regimes: a dominated-single-term law and a trimmed CLT leading to Gaussian behavior, depending on variance contributions. These results show that combinatorial regularity persists under randomness and open avenues for thresholds, infinite-structure questions, and extensions to other algebraic settings.

Abstract

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

TL;DR

The paper asks whether random subsets of the natural numbers retain additive and multiplicative structure typical of Hindman-type theorems. It develops and analyzes the Binomial model , proving that almost surely, FS-sets and FP-sets of every finite length appear, via disjoint-pattern constructions and the second Borel--Cantelli lemma. A novel link between Hindman’s partition theorem and the central limit theorem is established, yielding two limiting regimes: a dominated-single-term law and a trimmed CLT leading to Gaussian behavior, depending on variance contributions. These results show that combinatorial regularity persists under randomness and open avenues for thresholds, infinite-structure questions, and extensions to other algebraic settings.

Abstract

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of where each integer is included independently with probability , we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

Paper Structure

This paper contains 6 sections, 8 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results and Their Proofs
  4. Hindman’s Theorem and a Central Limit Phenomenon
  5. Corollaries of Main Theorems
  6. Conclusion and Open Problems

Key Result

Theorem 1

Fix $p\in(0,1)$. Let $A\subset\mathbb{N}$ be distributed as $\mathbb{N}_p$. Then with probability $1$, the following holds: For every integer $L\ge 1$ there exist integers $x_1<\cdots<x_L$ such that In other words, almost surely $A$ contains finite FS-sets of arbitrarily large finite length.

Theorems & Definitions (24)

  • Definition 1: Random Subset of $\mathbb{N}$
  • Remark 1
  • Definition 2: Finite Sumset
  • Example 1
  • Remark 2
  • Definition 3: Finite Product Set
  • Example 2
  • Remark 3
  • Theorem 1: Random Finite Sumsets
  • proof
  • ...and 14 more