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Minimal gap for higher dimensional sequences

This paper extends the concept of minimal gaps to higher-dimensional sequences by establishing bounds for specific sequences in terms of the cardinality of their associated difference sets.

Original authors: Tanmoy Bera

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tanmoy Bera

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 hosting a party where guests arrive one by one and sit in a long, circular hallway that is exactly one mile long. The hallway is marked with a ruler from 0 to 1.

The Basic Game: The "Minimal Gap"
In mathematics, a "minimal gap" is simply the smallest distance between any two guests sitting in that hallway. If you have 100 guests, you want to know: What is the tightest squeeze between any two people?

If you throw the guests into the hallway completely at random (like tossing darts at a board), math tells us that as the party gets huge, the smallest distance between two people will be incredibly tiny—roughly $1/N^2$ (where $N$ is the number of guests). It's like finding two grains of sand that are almost touching in a giant beach.

The Twist: The "Sequences"
Now, imagine the guests aren't arriving randomly. They are following a strict, pre-written rule (a "sequence").

Sequence A: Guest $n$ sits at position $n \times \alpha$ (where $\alpha$ is a secret number).
Sequence B: Guest $n$ sits at position $n^2 \times \alpha$ .

The paper asks: If we follow these strict rules, how small can the gap between two guests get? Does it behave like the random party, or do the rules force people to sit too far apart (or too close)?

The New Challenge: The "High-Dimensional" Party
The author, Tanmoy Bera, takes this game and moves it into higher dimensions. Instead of a single hallway (1D), imagine:

2D: A giant square floor. Guests sit at coordinates $(x, y)$ .
3D: A giant cube. Guests sit at $(x, y, z)$ .
d-D: A hyper-cube.

The "distance" is now the shortest path between two guests in this multi-dimensional space, wrapping around the edges (like a video game character who walks off the right edge and appears on the left).

The Main Discovery: The "Difference Set" is the Key
The paper's big insight is that the size of the smallest gap depends entirely on the difference set.

Think of the "difference set" as a list of all possible "steps" you can take between any two guests.

If Guest 5 is at position 5 and Guest 2 is at position 2, the "step" is 3.
If Guest 100 is at 100 and Guest 1 is at 1, the "step" is 99.

The paper proves that the size of the minimal gap is inversely related to the size of this list of steps.

Few unique steps? The guests are forced to cluster in specific ways, and the gaps might be larger.
Many unique steps? The guests are spread out more like the random party, and the gaps get very small.

The author provides formulas to calculate exactly how small these gaps will be for almost all secret numbers ( $\alpha$ ). The formulas involve the number of guests ( $N$ ) and the number of unique steps in the difference set ( $C_N$ ).

Specific Findings in the Paper

The "Vector" and "Linear" Games:
The paper looks at two types of high-dimensional rules:
- Vector Rule: Each guest $n$ has a unique ID for every dimension (e.g., Guest 1 is at $(1, 2, 3)$ , Guest 2 is at $(2, 4, 6)$ ).
- Linear Rule: The guest's position is a sum of their ID multiplied by different secret numbers (e.g., $n \times \alpha_1 + n \times \alpha_2$ ).
  The paper gives precise "upper and lower bounds" for the gaps in these scenarios. Essentially, it says: "If your list of steps is big enough, the gaps will be this small (or smaller)."
The "Van der Corput" Sequence (The Perfectly Organized Party):
The paper also looks at a very specific, famous way of arranging numbers called the Van der Corput sequence. This is like a party where guests are arranged with mathematical perfection to avoid clustering.
- The Result: For this specific sequence, the paper proves the minimal gap is never too small and never too large. It stays perfectly balanced, roughly $1/N$ . It's like a perfectly spaced row of chairs where no two are ever too close, no matter how many guests arrive.

The "Almost All" Caveat
The paper uses the phrase "for almost all $\alpha$ ." In plain English, this means: "If you pick a secret number $\alpha$ completely at random, these rules will hold true." There might be a few weird, specific numbers where the rules break, but if you pick one blindly, you are safe.

Summary
Tanmoy Bera's paper takes a classic math puzzle about the smallest distance between points and expands it into multi-dimensional space. The main takeaway is that the "tightness" of the squeeze between points in these complex sequences is determined by how many unique "steps" exist between the points. If the steps are diverse, the points scatter like a random crowd; if the steps are repetitive, the points behave differently. The paper provides the exact math to predict this behavior.

Technical Summary: Minimal Gap for Higher Dimensional Sequences

Problem Statement
The paper addresses the problem of quantifying the "minimal gap" for sequences in higher-dimensional spaces. For a sequence $(x_n)$ in $[0, 1)$ , the minimal gap is defined as $\delta_{\min}((x_n), N) = \min\{ \|x_n - x_m\| : 1 \le n \neq m \le N \}$ , where $\|x\|$ is the distance to the nearest integer. While the behavior of this statistic is well-understood for random sequences (asymptotic order $1/N^2$ ) and specific one-dimensional sequences of the form $\{a_n \alpha\}$ (where $\{ \cdot \}$ denotes the fractional part), this work extends the definition and analysis to higher-dimensional sequences.

Specifically, the paper investigates three types of sequences for a fixed dimension $d \ge 2$ :

Vector sequences: $\{ \mathbf{a}_n \boldsymbol{\alpha} \} = (\{a_n^{(1)}\alpha_1\}, \dots, \{a_n^{(d)}\alpha_d\})$ , where $\mathbf{a}_n = (a_n^{(1)}, \dots, a_n^{(d)})$ is a vector of distinct integers.
Component-wise sequences: $\{ \mathbf{a}_n \boldsymbol{\alpha} \}$ interpreted as a vector of fractional parts (similar to the first, but notation distinguishes the vector nature).
Linear forms: $\{ \mathbf{a}_n \cdot \boldsymbol{\alpha} \} = \{ \sum_{i=1}^d a_n^{(i)} \alpha_i \}$ , which results in a one-dimensional sequence.

The goal is to establish upper and lower bounds for these minimal gaps for almost all $\boldsymbol{\alpha} \in [0, 1]^d$ , expressed in terms of the cardinality of the difference sets of the underlying integer sequences and their additive energy.

Methodology
The proofs rely on metric Diophantine approximation techniques, specifically adapting methods from Rudnick [13] and Aistleitner, El-Baz, and Munsch [2] to the higher-dimensional setting.

Definitions and Norms: The author defines the nearest integer distance in $\mathbb{R}^d$ using the $\infty$ -norm ( $\| \cdot \|_\infty$ ). The minimal gap is analyzed via the difference sets $(A_N - A_N)^+$ , where $A_N$ is the set of the first $N$ terms of the sequence. The cardinality of this set, denoted $C_N$ , plays a central role in the bounds.
Borel-Cantelli Lemmas: For lower bounds, the first Borel-Cantelli lemma is applied to show that for almost all $\boldsymbol{\alpha}$ , the gap cannot be smaller than a specific function of $N$ and $C_N$ . For upper bounds, the second Borel-Cantelli lemma (or variants like the Chung-Erdős inequality) is used to demonstrate that the gap is smaller than a certain threshold for infinitely many $N$ (i.m.) or sufficiently large $N$ (s.l.).
Variance Method: To establish upper bounds for sufficiently large $N$ , the paper employs the variance method. It defines counting functions $D(N, M)$ and $\tilde{D}(N, M)$ that count pairs of indices where the distance is small. By estimating the variance of these functions and showing they are small relative to their means, the author proves that the counting functions are non-zero almost surely.
Additive Energy: The analysis incorporates the additive energy $E(A_N)$ , defined as the number of solutions to $a_{n_1} + a_{n_2} = a_{n_3} + a_{n_4}$ . The paper utilizes bounds on the $L^2$ norm of representation functions and joint additive energies for subsets of coordinates to refine the upper bounds.
Linear Forms and GCD Sums: For linear forms, the paper utilizes results on the distribution of linear forms modulo 1, including generalizations of the Duffin-Schaeffer conjecture and specific lemmas regarding the measure of sets defined by GCD conditions (following [2] and [5]).

Key Contributions and Results

Extension to Higher Dimensions: The primary contribution is the formal definition and analysis of minimal gaps for $d$ -dimensional sequences, extending previous one-dimensional results.
Bounds for Vector Sequences ( $\{ \mathbf{a}_n \boldsymbol{\alpha} \}$ ):
- Lower Bound: For almost all $\boldsymbol{\alpha}$ , $\delta_{\min}^\infty(\boldsymbol{\alpha}, N) \ge \frac{1}{C_N^{1/d} (\log N)^{1/d} (\log_2 N)^{1/d + \epsilon}}$ for sufficiently large $N$ .
- Upper Bound (i.m.): $\delta_{\min}^\infty(\boldsymbol{\alpha}, N) \le \frac{1}{(C_N \log N \log_2 N)^{1/d}}$ for infinitely many $N$ .
- Upper Bound (s.l.): Under conditions where the joint additive energy is small ( $E(A_N^{D'}) \ll N^{2+o(1)}$ for subsets of size $> d/2$ ), the bound improves to $\delta_{\min}^\infty(\boldsymbol{\alpha}, N) \le \frac{N^\delta}{N^{2/d}}$ .
Bounds for Linear Forms ( $\{ \mathbf{a}_n \cdot \boldsymbol{\alpha} \}$ ):
- Lower Bound: $\delta_{\min}(\boldsymbol{\alpha}, N) \ge \frac{1}{C_N \log N (\log_2 N)^{1+\epsilon}}$ .
- Upper Bound: The paper removes the sequence-dependent term found in previous works (like [12]), proving that for almost all $\boldsymbol{\alpha}$ , $\delta_{\min}(\boldsymbol{\alpha}, N) \le \frac{1}{C_N}$ for infinitely many $N$ .
- Refined Upper Bound: If the minimum additive energy of the component sequences is small, $\delta_{\min}(\boldsymbol{\alpha}, N) \le \frac{N^\delta}{N^2}$ .
Specific Sequences:
- Van der Corput Sequence: The minimal gap is bounded by $\frac{1}{bN} \le \delta_{\min} \le \frac{b}{N}$ for $b \ge 3$ , and $\delta_{\min} \le \frac{1}{N}$ for $b=2$ .
- Halton Sequence: The author conjectures (without proof) that the minimal gap for the $d$ -dimensional Halton sequence behaves as $\frac{1}{N^{1/d}}$ .

Significance and Claims
The paper claims to successfully generalize the theory of minimal gaps from one-dimensional sequences to higher dimensions. It demonstrates that the cardinality of the difference set ( $C_N$ ) is a critical parameter, analogous to the role of additive energy in one-dimensional cases.

The author notes that the results for linear forms improve upon recent work by Reghvim [12] by removing a sequence-dependent logarithmic factor from the upper bound. The paper also highlights that for specific sequences like $n^2 \alpha_2 + n \alpha_1$ , the minimal gap is of order $O(1/N^2)$ (up to logarithmic factors) for almost all $\alpha$ , consistent with random behavior, provided the additive energy is low.

The work is presented as a theoretical extension within metric Diophantine approximation, relying on established probabilistic and number-theoretic tools rather than proposing new experimental applications. The author modestly notes that while proofs are provided for integer sequences, the extension to positive real sequences with growth conditions is possible via similar arguments to those in [12].

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