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Precise Asymptotics and Exact Formulas for Tensor Product Energies of Fibonacci Lattices

This paper establishes precise asymptotic formulas and exact closed-form expressions for sums of tensor product energies involving Fibonacci lattices, demonstrating their linear growth behavior and connecting the underlying constants to the Dedekind zeta function over $\mathbb{Q}(\sqrt{5})$ .

Original authors: Melia Haase, Nicolas Nagel

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Melia Haase, Nicolas Nagel

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

The Big Picture: Arranging Guests at a Party

Imagine you are hosting a party in a square room (the "torus"). You have a specific number of guests to invite, and your goal is to seat them so that they are spread out as evenly as possible. You want to avoid "clumps" where everyone is crowded in one corner and "holes" where no one is sitting at all.

In mathematics, this is called discrepancy theory. The paper asks: What is the absolute best way to seat $N$ guests to make the room look perfectly uniform?

To measure "how good" a seating arrangement is, the authors use a concept called Energy. Think of the guests as tiny magnets that repel each other. If they are too close, the "energy" (or tension) is high. If they are perfectly spaced, the energy is low. The goal is to find the arrangement with the lowest possible energy.

The Special Guests: The Fibonacci Lattices

For a long time, mathematicians have suspected that a specific pattern of seating, based on Fibonacci numbers (1, 1, 2, 3, 5, 8, 13...), is the "champion" of uniformity. These are called Fibonacci Lattices.

If you arrange your guests using these numbers, the pattern looks incredibly smooth and balanced. The paper doesn't just assume this is true; it tries to prove why it works so well and exactly how good it is as the party gets bigger (as the number of guests $N$ grows).

The Mystery of the "Clumping"

When the authors looked closely at the math behind these Fibonacci arrangements, they noticed something strange and beautiful.

If you plot the "energy" of every single guest pair, the numbers don't just scatter randomly. They clump together in specific groups.

Imagine you are looking at a crowd of people. Most are standing randomly, but if you look closely, you see small groups of people standing in perfect lines.
In this math problem, the "lines" are formed by numbers that follow the Fibonacci rule (each number is the sum of the two before it).

The authors realized that the entire chaotic-looking sum of energies could be broken down into these neat, Fibonacci-like families. By grouping the guests this way, they could predict the total energy with incredible precision.

The Main Discovery: A Formula for the Future

The paper provides a "crystal ball" formula. If you know how many guests you have (specifically, if the number of guests is a Fibonacci number), you can calculate the total energy of the room using a simple equation:

$\text{Total Energy} \approx (\text{Constant } C) \times n + (\text{Constant } D) + \text{Tiny Error}$

$n$ : Represents the size of the Fibonacci number (how "big" the party is).
$C$ and $D$ : These are fixed numbers that depend on the specific rules of the "repulsion" between guests.
The Error: The difference between the prediction and reality is so small it vanishes almost instantly as the party gets bigger.

The authors didn't just guess these constants; they found a way to calculate them using a deep mathematical object called the Dedekind zeta function. You can think of this function as a "master key" that unlocks the hidden structure of numbers in the field of $\sqrt{5}$ (which is where Fibonacci numbers live).

The "Exact" Surprise

Usually, in math, when you deal with infinite sums and complex patterns, you only get an approximation. You say, "It's roughly this much."

However, for a specific, very common case (where the repulsion rule is squared, like distance squared), the authors found something magical: An Exact Formula.

They proved that for this specific case, the total energy isn't just close to a formula; it is the formula. It involves:

The number of guests ( $n$ ).
Fibonacci numbers ( $F_n$ ).
Lucas numbers (a cousin of Fibonacci numbers: 2, 1, 3, 4, 7...).
A tiny "wobble" term that flips sign depending on whether the party size is even or odd.

It's like finding a perfect, closed-door box for a puzzle that everyone thought required an open-ended, messy solution.

Summary in a Nutshell

The Problem: How to spread points out perfectly in a square to minimize "tension" (energy).
The Suspect: Fibonacci Lattices are the best arrangement.
The Method: The authors broke the problem down by noticing that the points naturally group into Fibonacci-like families.
The Result: They derived a precise formula that predicts the energy of these arrangements for any size.
The Bonus: For the most common case, they found a simple, exact equation (no approximations needed) that links the energy directly to Fibonacci and Lucas numbers.

In short, the paper takes a messy, infinite sum of trigonometric functions and reveals that underneath the chaos lies a perfectly ordered, predictable structure governed by the golden ratio.

Technical Summary: Precise Asymptotics and Exact Formulas for Tensor Product Energies of Fibonacci Lattices

Problem Statement
The paper investigates the asymptotic behavior of "tensor product energies" associated with Fibonacci lattices. These lattices, denoted $\Phi_n = \Lambda_{F_n, F_{n-1}}$ , are point sets in the two-dimensional unit torus $[0, 1)^2$ constructed using consecutive Fibonacci numbers $F_n$ . The study focuses on sums of the form:
$\frac{1}{F_n^\sigma} \sum_{m=1}^{F_n-1} \frac{f(m/F_n)}{|\sin(\pi m/F_n)|^\sigma} \frac{f(F_{n-1}m/F_n)}{|\sin(\pi F_{n-1}m/F_n)|^\sigma}$
where $\sigma > 1$ and $f$ is a symmetric, 1-periodic function. These sums arise in discrepancy theory and the analysis of quasi-Monte Carlo integration methods, where minimizing such energies corresponds to finding point sets with optimal uniformity (low discrepancy). While it is conjectured that Fibonacci lattices are global minimizers for certain parameters, the precise asymptotic structure of their energy, particularly the lower-order terms beyond the leading order, had not been rigorously established.

Methodology
The authors employ a combination of analytic number theory, combinatorial properties of the Fibonacci sequence, and the theory of the Wythoff array.

Decomposition via the Wythoff Array: The core technical innovation is the partitioning of the summation index set $\{m \in \mathbb{N} : m < F_n/2\}$ using the Wythoff array $(W_{i,k})$ . The rows of this array correspond to sequences satisfying the Fibonacci recurrence. This partition allows the authors to group terms in the sum that exhibit "clustering" behavior around specific values.
Dual Arrays and Approximation: The authors introduce a "dual array" $W^*_{i, n-k}$ to handle the interaction between the two sine terms in the sum. They derive precise approximations for the terms in the sum by analyzing the asymptotic behavior of $W_{i,k}$ and $W^*_{i, n-k}$ as $n \to \infty$ .
Error Analysis: Using the $\alpha$ -Hölder continuity of the function $f$ , the authors bound the error introduced by replacing the discrete sum terms with their asymptotic limits. They demonstrate that the error decays exponentially with respect to $n$ .
Connection to Dedekind Zeta Functions: The leading constant in the asymptotic expansion is identified as a series involving the invariants $\eta_i$ of the Wythoff array rows. The authors prove that this series is equivalent to the Dedekind zeta function of the algebraic number field $\mathbb{Q}(\sqrt{5})$ , denoted $\zeta_{\mathbb{Q}(\sqrt{5})}(\sigma)$ .
Reciprocity Laws for Exact Formulas: For specific cases where $f$ relates to Bernoulli polynomials (corresponding to integer $\sigma$ ), the authors utilize reciprocity laws for generalized Dedekind sums to derive exact closed-form expressions for the finite sums, rather than just asymptotic approximations.

Key Contributions and Results

Precise Asymptotic Expansion (Theorem 1.2): The paper establishes that for $\sigma > 1$ and a broad class of functions $f$ , the energy sum behaves asymptotically as:
$Cn + D + O\left(\frac{n^2}{\phi^{\min\{\alpha, \sigma-1\}n/2}}\right)$
where $\phi$ is the golden ratio. The constants $C$ and $D$ are explicitly defined via infinite series.
Explicit Constants:
- The leading coefficient $C$ is given by $C = \frac{2f(0)^2 5^{\sigma/2}}{\pi^{2\sigma}} \zeta_{\mathbb{Q}(\sqrt{5})}(\sigma)$ . This connects the energy of Fibonacci lattices directly to the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ .
- The constant term $D$ is expressed as a convergent series involving the error terms of the approximations and the invariants $\eta_i$ .
Exact Closed-Form Expressions: For the specific case $\sigma = 2$ (related to periodic $L^2$ -discrepancy) and $f(t)=1$ , the authors derive an exact formula for the sum:
$\sum_{m=1}^{F_n-1} \frac{1}{\sin^2(\pi m/F_n)\sin^2(\pi F_{n-1}m/F_n)} = \frac{4n}{75}F_n^2 - \frac{17}{225}L_n^2 - \frac{(-1)^n 2}{15} - \frac{1}{9}$
where $L_n$ are the Lucas numbers. Similar exact formulas are provided for $\sigma=4$ and $\sigma=6$ .
Numerical Verification: The paper provides numerical values for the constants $C$ and $D$ for $\sigma = 2, 4, 6$ , confirming the theoretical predictions.

Significance
The paper claims significance in providing the first rigorous analysis of the lower-order terms in the asymptotic expansion of tensor product energies for Fibonacci lattices. Prior to this work, only the leading order behavior was well-understood. By expressing the constants in terms of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ , the authors bridge the gap between the geometric properties of Fibonacci lattices and the arithmetic properties of the field $\mathbb{Q}(\sqrt{5})$ . Furthermore, the derivation of exact closed-form expressions for specific cases offers a deeper understanding of the structure of these sums, which are central to the study of discrepancy and numerical integration. The authors note that while the conjecture that Fibonacci lattices are global minimizers remains open, this precise control of the asymptotics is a necessary step toward resolving it.