🔢 mathematics

Multiplication Tables for Integers with Restricted Prime Factors

This paper generalizes Ford's 2008 results on multiplication tables by determining the order of magnitude for integers with prime factors restricted to a set of relative density $\delta$ that possess a divisor in a specific interval, while also identifying and characterizing a phase transition at the critical density $\delta = 1/\log 4$ .

Original authors: Jeremy Schlitt

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jeremy Schlitt

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite grid of numbers, like a massive multiplication table. If you fill this table with every possible number from 1 to $N$ , you get a chaotic mess of products. But what if you only allowed yourself to use numbers made from a specific "recipe" of ingredients?

That is the core puzzle Jeremy Schlitt solves in this paper. He looks at a special kind of multiplication table where the "ingredients" (prime numbers) are restricted. Instead of using all prime numbers (2, 3, 5, 7, 11...), he asks: What happens if we only use a specific subset of primes?

Here is the breakdown of his discovery, using simple analogies.

1. The Setup: The "Restricted Bakery"

Imagine a bakery that only sells bread made with specific types of flour.

The Standard Bakery: Uses every type of flour available. This is the classic multiplication table problem that mathematicians have studied for decades.
Schlitt's Bakery: Only uses a specific fraction ( $\delta$ ) of the available flours. Maybe they only use "red" flours, or flours that appear frequently in nature.

The question is: How many unique loaves of bread (products) can this bakery make?

If you mix two numbers from your restricted list, do you get a unique result every time? Or do you get a lot of duplicates (e.g., $2 \times 6 = 12$ and $3 \times 4 = 12$ )?

2. The "Divisor Detective" Game

To solve this, Schlitt doesn't just count the final products. He plays a game of "Divisor Detective."

He looks for numbers that have a "middle child" divisor. Imagine a number is a family. He asks: "Does this family have a member whose size is between $y$ and $2y$ ?"

If a number has a divisor in this specific range, it's a "good" number for the multiplication table.
The paper counts how many such numbers exist up to a certain limit ( $x$ ).

3. The Big Discovery: The "Tipping Point"

The most exciting part of the paper is a Phase Transition. Think of this like water freezing into ice. As you change the temperature (in this case, the density of primes, $\delta$ ), the behavior of the system suddenly changes.

Schlitt found a critical "tipping point" at a specific density: $\delta = 1 / \log(4)$ (which is roughly 0.72).

Scenario A: The "Sparse" Bakery ( $\delta < 0.72$ )
If you restrict your flour selection too much (less than 72% of the available primes), the bakery is very efficient. Almost every time you multiply two numbers, you get a unique result. There are very few duplicates. The multiplication table is "full" and distinct.
- Analogy: It's like a dance floor with few people. Everyone has plenty of space to move without bumping into others.
Scenario B: The "Crowded" Bakery ( $\delta > 0.72$ )
If you allow more types of flour (more than 72% of primes), the system becomes chaotic. Suddenly, you get massive amounts of duplicates. The number of unique products drops significantly compared to the total number of pairs you could make.
- Analogy: It's like a packed concert. Everyone is bumping into each other. Many people end up in the same spot (same product), so the number of distinct spots occupied is much lower than the number of people.

4. The "Random Walk" Metaphor

To prove this, Schlitt uses a concept from probability called a "Random Walk."
Imagine a drunk person walking down a street.

In the "Sparse" case, the person is walking on a wide sidewalk. They can wander a bit, but they rarely hit a wall.
In the "Crowded" case, the person is walking through a narrow alley with walls on both sides. They are forced to stay very close to the center line.

Schlitt proved that when the density of primes crosses that 0.72 threshold, the "drunk walker" (the mathematical structure of the numbers) suddenly hits a "barrier" that forces them to behave differently. This barrier is what causes the explosion of duplicate products.

5. Why Does This Matter?

This isn't just about counting numbers. It helps mathematicians understand:

Structure in Chaos: How order emerges from random-looking sets of numbers.
Cryptography: Many encryption systems rely on the difficulty of factoring numbers. Understanding how divisors are distributed helps us know how "safe" or "predictable" certain number sets are.
The "Multiplication Table Constant": Schlitt generalized a famous constant discovered by Kevin Ford in 2008. He showed that this constant isn't just a fixed number; it changes shape depending on how "rich" the set of prime numbers is.

Summary

Jeremy Schlitt took a complex math problem about multiplication tables and showed that the density of the ingredients matters more than we thought.

Low Density: You get a clean, unique list of products.
High Density: You get a messy list full of duplicates.
The Switch: There is a precise mathematical "switch" (at $\approx 0.72$ ) where the system flips from one behavior to the other.

He did this by inventing new ways to visualize numbers as geometric shapes and using probability to predict how "crowded" the number line gets. It's a beautiful example of how changing a single rule (which primes you use) completely reshapes the landscape of mathematics.

1. Problem Statement

The paper addresses a generalization of Erdős' Multiplication Table Problem, which concerns the number of distinct integers in an $N \times N$ multiplication table. Specifically, it investigates the asymptotic behavior of the number of distinct products $ab$ where $a$ and $b$ are restricted to a set of integers $S_Q$ whose prime factors belong to a specific subset of primes $Q$ .

Let $Q$ be a set of primes with relative density $\delta \in (0, 1]$ . This means the counting function of $Q$ satisfies:
$\left| \#(Q \cap [1, x]) - \frac{\delta x}{\log x} \right| \leq \frac{\kappa x}{(\log x)^2}$
The set of interest is $S_Q = \{n \in \mathbb{N} : p|n \implies p \in Q\}$ . The author defines:

$H_Q(x, y, z)$ : The number of integers $n \in S_Q \cap [1, x]$ having a divisor in the interval $(y, z]$ .
$A_Q(N)$ : The number of distinct integers in the multiplication table restricted to $S_Q \cap [1, N]$ , i.e., $|\{ab : a, b \in S_Q \cap [1, N]\}|$ .

The primary goal is to determine the order of magnitude of $H_Q(x, y, 2y)$ and $A_Q(N)$ for all $\delta \in (0, 1]$ , generalizing Ford's seminal 2008 result for the case $\delta = 1$ (all primes).

2. Methodology

The proof follows the probabilistic and combinatorial framework established by Kevin Ford (2008) but introduces significant refinements to handle the density parameter $\delta$ .

A. Reduction to Poisson-Type Sums

The problem is reduced to estimating sums involving the function $L(a)$ , which measures the "propinquity" (closeness) of the logarithms of the divisors of $a$ .
$L(a) = \text{meas}\left( \bigcup_{d|a} (\log d - \log 2, \log d] \right)$
The author establishes that $H_Q(x, y, 2y)$ is asymptotically equivalent to a weighted sum of $L(a)/a$ over square-free integers $a \in S_Q$ . This sum is then approximated by a Poisson-type sum:
$\sum_{1 \leq k \leq v} \frac{\lambda^k}{k!} \cdot \frac{v - k + 1}{v}$
where $\lambda = 2\delta \log \log y$ and $v = \lfloor \frac{1}{\log 2} \log \log y \rfloor$ .

B. Geometric Interpretation and Barrier Conditions

A key innovation in this paper is the geometric interpretation of the lower bound.

Order Statistics: The problem is mapped to the volume of a region defined by uniform order statistics $\xi_1, \dots, \xi_k$ on $[0, 1]$ .
Strong Barrier Conditions: Unlike previous works (Ford, Koukoulopoulos) that used implicit algebraic constraints, Schlitt explicitly defines a geometric region $Y_k(\tilde{v}, C)$ with strong barrier conditions. These conditions require the normalized order statistics to repel from their mean values by a margin of roughly $(\min\{i, k-i\})^{1/7}$ .
Probabilistic Insight: The author leverages a known phenomenon in probability: random walks conditioned to satisfy a weak barrier constraint will, with high probability, satisfy a much stronger barrier condition. This allows for a precise calculation of the volume of the geometric region, which corresponds to the lower bound of the divisor sum.

C. Phase Transition Analysis

The behavior of the Poisson-type sum changes drastically depending on the relationship between $\lambda$ and $v$ . The critical point is identified as $\delta = 1/\log 4$ .

Case $\delta < 1/\log 4$ : The sum is dominated by terms where $k \approx \lambda$ . The behavior is governed by the tail of the Poisson distribution.
Case $\delta > 1/\log 4$ : The sum is dominated by the boundary term $k=v$ .
Case $\delta = 1/\log 4$ : A phase transition occurs where the behavior shifts, requiring a specific analysis of the transition region using saddle-point methods and refined estimates of the Poisson partial sums.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

The paper establishes the order of magnitude for $H_Q(x, y, 2y)$ uniformly for $\delta \geq c > 0$ :
$H_Q(x, y, 2y) \asymp_{\kappa, c} \frac{x}{(\log x)^{1-\delta}} \cdot \frac{1}{(\log y)^{G(\delta)}} \cdot E(y; \delta)$
Where:

$G(\delta)$ is the exponent of the logarithmic factor:
$G(\delta) = \begin{cases} 1 - \delta & \text{if } \delta \leq \frac{1}{\log 4} \\ \delta - \frac{1 + \log(\delta \log 2)}{\log 2} & \text{if } \delta \geq \frac{1}{\log 4} \end{cases}$
Note that $G(\delta)$ is differentiable at the critical point $\delta = 1/\log 4$ .
$E(y; \delta)$ captures the transition behavior near the critical point:
- For $\delta \leq 1/\log 4$ , it involves a term like $\max(1/\log 4 - \delta, 1/\sqrt{\log \log y})$ .
- For $\delta > 1/\log 4$ , it involves a term like $(\delta - 1/\log 4)^2$ .

Corollary on Multiplication Tables (Corollary 1.3)

Applying the main theorem to the multiplication table count $A_Q(N)$ :
$A_Q(N) \asymp_{c, \kappa} \frac{N^2}{(\log N)^{G(\delta) - \delta + 1}} E(N; \delta)$

Combinatorial Interpretation (Equation 1.6)

The paper provides a sharp dichotomy for the size of the product set $S_Q(N) \cdot S_Q(N)$ :

If $\delta < 1/\log 4$ : The product set is "large," meaning $|S_Q(N) \cdot S_Q(N)| \asymp |S_Q(N)|^2$ . Almost all products are distinct.
If $\delta \geq 1/\log 4$ : The product set is "small," meaning $|S_Q(N) \cdot S_Q(N)| = o(|S_Q(N)|^2)$ . There is significant overlap in the products.

4. Significance

Generalization of Ford's Work: This paper successfully extends the 2008 results of Kevin Ford (who solved the case for all primes, $\delta=1$ ) to the entire range of prime densities $\delta \in (0, 1]$ .
Discovery of a Phase Transition: The identification of $\delta = 1/\log 4$ as a critical threshold is a major theoretical breakthrough. It reveals a fundamental shift in the arithmetic structure of integers with restricted prime factors, separating regimes where products are distinct from regimes where they collide.
Methodological Innovation: The explicit use of strong barrier conditions for order statistics provides a clearer geometric intuition for the lower bounds than previous implicit methods. This technique connects number theory more deeply with probabilistic limit theorems (specifically Smirnov statistics and random walks).
Connections to Other Fields: The results have implications for the study of random multiplicative functions (as noted in the connection to Soundararajan and Xu) and extremal set theory (connecting to Ford's work on product sets).

In summary, Schlitt's paper resolves the asymptotic behavior of multiplication tables for integers with restricted prime factors, uncovering a rich phase transition structure that depends on the density of the allowed prime factors.