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Primitive inhomogeneous approximation for fixed non-singular frequencies

This paper establishes a high-dimensional primitive inhomogeneous Diophantine approximation theorem for fixed non-singular simultaneous frequencies, demonstrating that the result holds for an explicit frequency class of full Lebesgue measure.

Original authors: Xueyin Wang

Published 2026-06-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xueyin Wang

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 standing in a giant, multi-dimensional room (a mathematical space called a torus). You have a specific, fixed walking pattern defined by a set of numbers called a "frequency" ( $\alpha$ ). Every time you take a step, you move according to this pattern.

The paper by Xueyin Wang is about a game of "closest approach" played in this room. Here is the breakdown of the game, the rules, and the surprising result, explained through simple analogies.

The Game: "The Primitive Hunter"

The Setup:
Imagine you are trying to catch a moving target.

The Walker: You take steps labeled $n = 1, 2, 3, \dots$ . Your position after step $n$ is determined by your fixed walking pattern ( $\alpha$ ).
The Target: There is a specific spot in the room you want to get close to (let's call it $\gamma$ ).
The Goal: You want to find a step number $n$ where your position is incredibly close to the target.

The Twist (The "Primitive" Rule):
In the past, mathematicians allowed you to use any step number $n$ . But this paper adds a strict rule: You can only use "Primitive" steps.

Think of a step number $n$ as a team of people. If $n=6$ , the team is made of numbers $(6, m_1, m_2, \dots)$ .
A "Primitive" team is one where the members have no common factor other than 1. They are "coprime."
If the team shares a secret handshake (a common factor like 2 or 3), that step is disqualified. You must wait for a "pure" team to take the step.

The Challenge:
The question is: If you are forced to skip all the "impure" steps, can you still get arbitrarily close to your target? And how fast do you have to get close as you take more steps?

The "Bad" Walkers vs. The "Good" Walkers

The author focuses on a specific type of walking pattern called "Non-Singular."

The Bad Walkers (Singular): Imagine a walker who, no matter how long they walk, always stays stuck in a narrow hallway or a flat sheet within the room. They never truly explore the whole space. These are the "bad" frequencies.
The Good Walkers (Non-Singular): These walkers are chaotic in a good way. They don't get stuck in narrow hallways. They spread out evenly across the entire room. The paper proves that almost all random walking patterns fall into this "Good" category.

The Big Discovery

The paper proves a very specific result about these "Good Walkers":

Even if you are forced to skip all the "impure" steps, you can still get as close to your target as you want.

Here is the magic part:
In a room with $d$ dimensions, the "closeness" you can achieve is related to the size of the room and the number of steps.

If you take $n$ steps, the distance you can get to the target shrinks roughly by a factor of $1/n^{1/d}$ .
The paper shows that even with the strict "Primitive" rule (skipping impure steps), you do not lose any ground. You can still reach that same level of closeness.

The Analogy of the "Shrinking Target"

Imagine the target is a tiny dot that gets smaller and smaller as you take more steps.

Without the Primitive Rule: It's like throwing darts at a shrinking bullseye. You know you will hit it eventually.
With the Primitive Rule: It's like being told, "You can only throw darts on Tuesdays."
- The paper asks: "If you only throw on Tuesdays, do you still hit the shrinking bullseye?"
- The Answer: Yes! As long as your walking pattern is a "Good Walker" (Non-Singular), the fact that you are skipping days (steps) doesn't stop you from hitting the target. The "Tuesday-only" constraint doesn't make the target harder to hit in the long run.

How They Proved It

The author used two main tools to solve this puzzle:

The Sieve (Filtering the Steps):
Imagine you have a giant bucket of step numbers. You want to keep only the "Primitive" ones. The author used a mathematical "sieve" (based on an old technique called the Möbius function) to filter out the impure steps. They showed that even after filtering, the remaining steps are still spread out evenly enough to do the job.
The "Good Scales" (Timing the Hits):
The author didn't look at every single step. Instead, they looked at specific "good moments" (scales) where the walker is known to be well-distributed. They built a safety net of small boxes around these good moments. They proved that if you keep adding these boxes over time, they eventually cover the entire room. This means no matter where your target is, it will eventually fall inside one of these boxes.

The Bottom Line

The paper solves a high-dimensional puzzle about how numbers interact. It confirms that for almost every "good" walking pattern, the restriction of using only "primitive" (coprime) steps does not ruin your ability to get infinitely close to a target. The "primitive" constraint is a hurdle, but it is not a wall; the walker can still jump over it and reach the goal.

In short: If you are a good walker, skipping the "impure" steps won't stop you from finding your way to any spot in the room, no matter how small that spot gets.

Technical Summary: Primitive Inhomogeneous Approximation for Fixed Non-Singular Frequencies

Problem Statement
This paper addresses the problem of primitive inhomogeneous Diophantine approximation in higher dimensions ( $d \ge 2$ ). Specifically, it investigates the behavior of the approximation error for a fixed frequency vector $\alpha \in \mathbb{R}^d$ and a target vector $\gamma \in \mathbb{R}^d$ . The goal is to determine whether the quantity
$\liminf_{n \to \infty} n^{1/d} \min_{\substack{m \in \mathbb{Z}^d \\ \gcd(n, m_1, \dots, m_d)=1}} \|\gamma - n\alpha + m\|_\infty$
equals zero for Lebesgue almost every $\gamma$ .

The constraint $\gcd(n, m_1, \dots, m_d) = 1$ imposes a "primitive" condition on the integer vector $(n, m_1, \dots, m_d)$ , distinguishing this from standard inhomogeneous approximation. While the one-dimensional case ( $d=1$ ) was resolved by Jitomirskaya and Liu [JL19] for every irrational $\alpha$ , the higher-dimensional analogue requires a more nuanced hypothesis on $\alpha$ . Rational independence alone is insufficient, as demonstrated by counterexamples from Galatolo and Peterlongo [GP10] regarding singular vectors.

Methodology
The proof is structured around two primary components: a primitive discrepancy estimate and a shrinking-target construction.

Primitive Discrepancy Estimates (Section 3):
The authors establish an asymptotic count for the number of integers $n \le N$ such that the orbit point $\{n\alpha\}$ falls into a Jordan measurable set $A$ and the vector $(n, \lfloor n\alpha_1 \rfloor - b_1, \dots, \lfloor n\alpha_d \rfloor - b_d)$ is primitive.
- Sieve Method: An elementary sieve is employed using the Möbius function $\mu(r)$ to filter out non-primitive vectors.
- Weyl Equidistribution: For fixed square-free $r$ , the condition that $r$ divides the components is translated into congruence conditions on the fractional parts $\{n\alpha\}$ . The uniform distribution of $\{n\alpha\}$ (guaranteed by the linear independence of $1, \alpha_1, \dots, \alpha_d$ over $\mathbb{Q}$ ) allows for the estimation of these counts.
- Error Control: The contribution of large prime divisors is controlled using a packing estimate derived from the "good scale" condition (1.1). This condition ensures that the orbit points $\{n\alpha\}$ are sufficiently separated, preventing clustering that would violate the discrepancy bounds.
Shrinking-Target Construction (Section 4):
To prove the result for almost every $\gamma$ , the authors construct a full-measure set of admissible targets using a second-moment Borel–Cantelli lemma.
- Target Sets: For a sequence of "good scales" $N_j$ (where the separation condition holds), small boxes of side length $r_j \sim N_j^{-1/d}$ are placed around the primitive orbit points $\{n\alpha\}$ .
- Quasi-Independence: A key technical step involves proving that the intersection of target sets at different scales behaves asymptotically like the product of their measures. This quasi-independence is established by analyzing the boundary effects of the boxes and utilizing the separation property of the orbit.
- Measure Argument: By selecting a subsequence of scales and applying the second-moment lemma, the authors show that the union of these shrinking targets covers almost every point in the torus.

Key Results
The main theorem (Theorem 1.1) states that for any dimension $d \ge 2$ and any frequency $\alpha$ belonging to the explicit class $\Lambda_d$ (defined by linear independence over $\mathbb{Q}$ and a non-singularity condition involving a lower bound on $\min_{1 \le n \le N} \|n\alpha\|$ ), the primitive inhomogeneous approximation holds for Lebesgue almost every $\gamma \in \mathbb{R}^d$ .

Specifically, for $\alpha \in \Lambda_d$ :
$\liminf_{n \to \infty} n^{1/d} \min_{\substack{m \in \mathbb{Z}^d \\ \gcd(n, m_1, \dots, m_d)=1}} \|\gamma - n\alpha + m\|_\infty = 0$
The class $\Lambda_d$ is shown to have full Lebesgue measure.

Significance and Claims
The paper claims two primary novelties:

Fixed-Frequency Result: Unlike previous metric results (e.g., Dani, Laurent, and Nogueira [DLN15]) which establish the result for almost every pair $(\alpha, \gamma)$ , this work provides a result for a fixed prescribed $\alpha$ , provided $\alpha$ is non-singular. This addresses the complementary problem raised by Allen and Ramírez [AR25].
Preservation of Critical Scale: The result demonstrates that the primitive restriction (the coprimality condition) does not destroy the critical shrinking-target scale of $n^{-1/d}$ . This scale is optimal for simultaneous approximation in dimension $d$ .

The authors position this work as a natural extension of the one-dimensional theory of Jitomirskaya and Liu [JL19] and the non-primitive higher-dimensional work of Shapira [Sha13]. The non-singularity condition is identified as the correct hypothesis for fixed-frequency theorems in higher dimensions, replacing simple rational independence.