Multiplication Tables for Integers with Restricted Prime Factors

TL;DR

This paper analyzes the distribution of integers with prime factors in a restricted set, establishing their count's order of magnitude and revealing a phase transition at a critical density, generalizing previous results.

Contribution

It generalizes Ford's 2008 results by determining the count of integers with restricted prime factors and identifying a phase transition at a specific prime density.

Findings

Established the order of magnitude for integers with prime factors in a set of density δ.

Identified a phase transition at δ = 1/ log 4 with explicit behavior.

Extended previous work from δ=1 to all δ in (0,1].

Abstract

Let $Q$ be a set of primes with relative density $\delta$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $\delta \in (0,1]$. This generalizes the case $\delta = 1$ from the 2008 work of Ford. We also show that there is a phase transition at the critical point $\delta = 1/\log 4$, for which we explicitly determine the behaviour.