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A Note on Diophantine Approximation with Restricted Denominators

This paper introduces a specific density notion for subsets of natural numbers to establish a restricted analog of Dirichlet's theorem on rational approximations to irrational numbers where denominators are limited to those subsets.

Original authors: Chance Sanford

Published 2026-06-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Chance Sanford

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 are trying to hit a bullseye on a dartboard that represents an irrational number (a number like $\pi$ or $\sqrt{2}$ that goes on forever without repeating). You want to throw a dart (a fraction, like $3/1$ or $22/7$ ) that lands as close to the bullseye as possible.

In the world of math, there is a famous rule by a mathematician named Dirichlet. It says: "No matter how weird your target number is, you can always find a fraction that gets incredibly close to it. In fact, the closer you get, the better your chances of finding an even closer fraction, provided you are allowed to use any whole number as the bottom part of your fraction (the denominator)."

The Problem: The "Restricted" Dartboard

The author of this paper, Chance Sanford, asks a new question: What if you aren't allowed to use any number for the bottom of your fraction?

Imagine someone puts up a fence around your dartboard. They say, "You can only throw darts at numbers that are in this specific list."

Maybe the list only contains prime numbers (2, 3, 5, 7...).
Maybe it only contains perfect squares (1, 4, 9, 16...).
Maybe it's a weird, custom list.

If the list is too "sparse" (too empty), you might never get close enough to the bullseye. But if the list is "dense" enough (full of numbers), you should still be able to hit the target, even if the rules are stricter.

The Solution: Measuring "Crowdedness"

Sanford introduces a new way to measure how "crowded" a list of numbers is. He calls this Diophantine density.

Think of it like a party:

High Density: The party is packed. If you look at a small section of the room, there are tons of people.
Low Density: The party is empty. You might have to walk a long way to find another person.

Sanford proves a simple but powerful rule: If your list of allowed numbers is "crowded enough" (has a high enough density), you can still find fractions that get very close to your irrational target number.

The "closer" you can get depends on how crowded the list is.

If the list is super crowded (like all natural numbers), you get the best possible result (Dirichlet's original rule).
If the list is less crowded, you still get a good result, but the math says the fraction won't be quite as perfect as the unrestricted version.

The "Complement" Trick

The paper also gives a clever trick for building these lists. Instead of trying to build a "good" list from scratch, you can start with a "bad" list (one that is too sparse) and take everything NOT in it.

For example:

Imagine a list of "Piatetski-Shapiro numbers." These are numbers generated by a specific, slightly weird formula (like rounding off $n^{1.5}$ ).
This specific list is actually quite "thin" (not many of them exist).
Sanford shows that if you take all the natural numbers and remove this thin list, the remaining numbers are "thick" enough to be a great list for hitting the bullseye.

The "Prime Number" Reality Check

The paper ends with a reality check about Prime Numbers.
Many people wonder: "Can we hit the bullseye using only prime numbers as our denominators?"

Sanford uses his new "crowdedness" meter to check the primes. He finds that while there are a lot of primes, they are actually too sparse to fit his specific definition of "dense enough" for his proof.

The Catch: This doesn't mean it's impossible to use primes (other mathematicians have proven it is possible, but they used very different, much harder tools).
The Limit: It just means that Sanford's specific, simpler "crowdedness" method isn't strong enough to prove it for primes. His method works best for lists that are "thick" but not as thin as the primes.

Summary

In short, this paper says:

If you restrict your math problems to a specific list of numbers, you can still get very good answers.
The quality of the answer depends on how "full" that list is.
We can predict how good the answer will be by measuring the "crowdedness" of the list.
This method works well for many lists (like numbers that aren't perfect squares), but it's too simple to solve the hardest puzzles (like using only prime numbers).

Problem Statement
This note addresses the problem of Diophantine approximation when the denominators of the approximating rational numbers are restricted to a specific subset $A$ of the natural numbers $\mathbb{N}$ . While Dirichlet's approximation theorem guarantees that for any irrational $\xi$ , there are infinitely many coprime pairs $(p, q)$ satisfying $|\xi - p/q| < 1/q^2$ , the author investigates whether a similar bound can be achieved when $q$ is constrained to lie within a subset $A \subseteq \mathbb{N}$ . The paper acknowledges that if $A$ is too sparse, the $q^{-2}$ bound is likely unattainable. Previous research in this area, such as Matomäki's work on denominators restricted to primes, has relied heavily on analytic number theory. This paper seeks to approach the problem through a combinatorial lens by introducing a specific notion of density.

Methodology
The author introduces a combinatorial definition of "Diophantine density" for subsets of $\mathbb{N}$ and utilizes elementary number theory and properties of continued fractions to establish results.

Definition of Diophantine Density: A set $A \subseteq \mathbb{N}$ is defined to have a Diophantine density $\delta \in [0, 1]$ if there exists a constant $C > 0$ such that for every sufficiently large coprime pair $(p, q)$ , the minimum distance from $vp/q$ to the nearest integer, minimized over $v \in A$ with $v \le q$ , is bounded by $C/q^\delta$ .
- Remark: The author proves that any set with positive Diophantine density must be infinite.
Combinatorial Construction (Theorem 4): The paper establishes a method to construct sets with specific Diophantine densities by analyzing the complement of sets with controlled growth. Specifically, if a set $A$ has a counting function $A(n) \ll n^\delta$ , then its complement $\mathbb{N} \setminus A$ possesses a Diophantine density of $1 - \delta$ . This is proven by analyzing the distribution of the multiset $\{ \|kp/q\| : 1 \le k \le q \}$ and showing that removing elements corresponding to $A$ leaves a minimum element bounded by the density of the removed set.
Application to Approximation (Theorem 3): Using the definition of density, the author derives a restricted analog of Dirichlet's theorem. If $A$ has Diophantine density $\delta > 0$ , then for any irrational $\xi$ and $\epsilon > 0$ , there exist infinitely many coprime pairs $(u, v) \in \mathbb{Z} \times A$ such that $|\xi - u/v| < 1/v^{1+\delta-\epsilon}$ . The proof relies on the convergents of $\xi$ and the triangle inequality, demonstrating that the density condition ensures the existence of infinitely many distinct coprime pairs.

Key Results

Theorem 3 (Restricted Dirichlet): If a subset $A \subseteq \mathbb{N}$ has Diophantine density $\delta > 0$ , then for any irrational $\xi$ , there are infinitely many approximations with denominators in $A$ satisfying $|\xi - u/v| < v^{-(1+\delta-\epsilon)}$ .
Theorem 4 (Density of Complements): If a set $A$ satisfies $A(n) \ll n^\delta$ , then the complement $\mathbb{N} \setminus A$ has Diophantine density $1 - \delta$ .
Theorem 5 (Piatetski-Shapiro Sequences): Applying the previous theorems to Piatetski-Shapiro sequences $PSc = \{\lfloor n^c \rfloor\}_{n \ge 1}$ $P S c = {⌊ n^{c} ⌋}_{n \geq 1}$ (where $c > 1$ $c > 1$ ), the author shows that the set of integers avoiding these sequences allows for approximations with an error bound of $|\xi - p/q| < q^{-(2 - 1/c - \epsilon)}$ $∣ ξ - p / q ∣ < q^{- (2 - 1/ c - ϵ)}$ .
- Example: For $c=2$ (avoiding perfect squares), the bound is $q^{-(3/2 - \epsilon)}$ .

Significance and Limitations
The paper claims significance in providing a "combinatorial approach" to restricted Diophantine approximation, contrasting with the "analytic number theory" methods used in prior work (e.g., Vinogradov, Matomäki). It establishes a direct link between the growth rate of a set's counting function and the quality of Diophantine approximations possible using its complement.

However, the author is modest regarding the scope of these results. The paper explicitly states that the methods are "too coarse" to recover stronger results known for specific sparse sets, such as the set of prime numbers. By applying Theorem 4, the author notes that the set of primes has a Diophantine density of 0 (since the set of composites has density 1), rendering Theorem 3 inapplicable to primes. Thus, the work is presented as a general framework for "sufficiently dense" subsets rather than a solution to the most difficult cases of sparse sets.

The Problem: The "Restricted" Dartboard

The Solution: Measuring "Crowdedness"

The "Complement" Trick

The "Prime Number" Reality Check

Summary

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