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On products of sets of natural density one

Sandro Bettin, Matteo Bordignon, Alessandro Fazzari

TL;DR

The paper tackles the rate at which the product of density-1 sets approaches density 1 by constructing a density-1 set A whose self-product A·A has a notably large complement, thereby bounding the optimal rate function ψ(a) from above. The authors build A via a discretization over dyadic prime-intervals with a large-prime divisor criterion Ω^*(m) and a continuous parameterization, enabling explicit bounds on R_x(A) and a quantified Omega-lower bound for R_x(A·A). They develop Poissonian and Halász-type estimates for restricted prime-divisor counts, enabling precise density calculations and optimization over parameters to demonstrate quadratic decay of ψ(a) as a→0^+. The results sharpen understanding of how quickly density-1 product sets can fail to cover initial segments of the integers and provide a concrete construction illustrating near-optimal behavior for small a.

Abstract

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers $A$ and $B$ have natural density $1$, then their product set $A \cdot B := \{ab : a \in A, b \in B\}$ also has natural density $1$. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set $A$ of density 1 such that $A\cdot A$ has a ''large'' complement.

On products of sets of natural density one

TL;DR

The paper tackles the rate at which the product of density-1 sets approaches density 1 by constructing a density-1 set A whose self-product A·A has a notably large complement, thereby bounding the optimal rate function ψ(a) from above. The authors build A via a discretization over dyadic prime-intervals with a large-prime divisor criterion Ω^*(m) and a continuous parameterization, enabling explicit bounds on R_x(A) and a quantified Omega-lower bound for R_x(A·A). They develop Poissonian and Halász-type estimates for restricted prime-divisor counts, enabling precise density calculations and optimization over parameters to demonstrate quadratic decay of ψ(a) as a→0^+. The results sharpen understanding of how quickly density-1 product sets can fail to cover initial segments of the integers and provide a concrete construction illustrating near-optimal behavior for small a.

Abstract

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers and have natural density , then their product set also has natural density . They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set of density 1 such that has a ''large'' complement.

Paper Structure

This paper contains 9 sections, 5 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sketch of the argument
  3. Notations
  4. Lemmata
  5. Proof of Theorem \ref{['mathe']}
  6. The density of A and A · A
  7. Proof of Equation
  8. Proof of Equation
  9. Proof of Equation

Key Result

Theorem 1.1

For $a\in(0,1/4)$, $B\in [0,\phi^{-1}(4a)]$, let where $\phi:[0,1]\to[0,1]$ is defined by Then, defining we have Moreover, one has

Figures (1)

  • Figure 1: The functions $\frac{a^2}{1+a}$ (green), $K(a)$ (blue) and $a$ (orange) for $0\leq a\leq0.15$. The function $\psi$ lies between the green curve and the minimum between the blue and the orange curves.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof