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arxiv: 2605.20895 · v1 · pith:ITOMAPVUnew · submitted 2026-05-20 · 🧮 math.NT · cs.NA· math.MG· math.NA

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

Melia Haase , Nicolas Nagel This is my paper

Pith reviewed 2026-05-21 02:42 UTC · model grok-4.3

classification 🧮 math.NT cs.NAmath.MGmath.NA
keywords Fibonacci numbersDedekind zeta functionasymptotics of sumstensor product energiesdiscrepancy theoryenergy minimizationexact formulasQ(sqrt 5)
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The pith

Sums appearing in Fibonacci lattice energies grow linearly in the index n plus a constant, with exponentially small remainder, and admit exact closed forms in special cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies sums that arise when reformulating numerical integration or discrepancy problems as energy minimization on points spaced according to Fibonacci numbers. For exponents sigma greater than one and sufficiently regular test functions f, these sums are shown to equal C times n plus a constant D plus a term that decays exponentially in n. The constants C and D themselves are expressed as infinite series built from the Dedekind zeta function of the quadratic field Q of square root five. In one explicit special case the sum is proved to equal a simple quadratic polynomial in the Fibonacci numbers F_n and F_{2n} together with a small periodic correction.

Core claim

The sum (1 over F_n to the sigma) times the double sum over m of f(m over F_n) over absolute sin of pi m over F_n to the sigma, times the analogous term with F_{n-1} m, behaves for large n like C n plus D plus an error bounded by a geometric term (1 minus epsilon) to the n. The leading coefficients C and D are given by explicit series involving values of the Dedekind zeta function attached to Q(sqrt 5). When f is identically one and sigma equals two the sum collapses exactly to (4n over 75) F_{2n} minus (17 over 225) F_n squared minus (-1)^n times (2 over 15) minus one over nine.

What carries the argument

Decomposition of the sum that exploits the linear recurrence of the Fibonacci sequence together with the algebraic integers in the ring of Q(sqrt 5) to isolate the linear growth and to express the constant term via zeta-series.

If this is right

  • The energy of the Fibonacci lattice point set therefore increases exactly linearly with the Fibonacci index.
  • The leading coefficient C can be evaluated to arbitrary precision by summing the associated zeta series.
  • Exact polynomial expressions in Fibonacci numbers become available for certain choices of f and sigma.
  • The exponentially small error term implies that truncation after the linear and constant terms already gives high-accuracy approximations for moderate n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same recurrence-plus-zeta technique may extend to other quadratic irrational rotations or to higher-dimensional product lattices built from Fibonacci sequences.
  • Because the error decays exponentially, these formulas could supply rigorous a-priori error bounds for quasi-Monte Carlo integration rules that use Fibonacci points.
  • The appearance of the Dedekind zeta function suggests a hidden arithmetic symmetry that might be visible in the distribution of the sine products themselves.

Load-bearing premise

The test functions f must be regular enough that the sum splits cleanly according to the Fibonacci recurrence without extra boundary or approximation errors.

What would settle it

Compute the sum numerically for n around 30 with sigma equal to 2 and f equal to one; the exact closed-form formula must hold to machine precision, and for general regular f the remainder after subtracting the predicted linear term must fall below 10 to the minus 5.

Figures

Figures reproduced from arXiv: 2605.20895 by Melia Haase, Nicolas Nagel.

Figure 1.1
Figure 1.1. Figure 1.1: Three point sets X1, X2, X3 (from left to right) of size N = 45 with e2,6(X1) = 0.037156 . . . , e2,6(X2) = 0.006865 . . . and e2,6(X3) = 0.001879 . . . ((σ, p) = (2, 6) corresponding to the notion of periodic L2-discrepancy [32, 33]). Observe how point sets with high and low density regions (clusters and holes respectively) corre￾spond to bigger eσ,p(X). 1.1. Uniformity via energy minimization Discrepan… view at source ↗
Figure 1
Figure 1. Figure 1: illustrates how [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Fibonacci lattices of size 3, 5, 8, 13 and 21. have more explicitly K2s,p(t) = 1 + p (−1)s−1 (2s)! B2s(t) with the (periodized) Bernoulli polynomials B2s(t) of degree 2s, in general given via X∞ n=0 Bn(t) x n n! = x e x − 1 e tx for t ∈ [0, 1). The next section introduces a class of point sets which will yield candi￾dates for minimizers of eσ,p(X). 1.2. Lattices Given two numbers N ∈ N, h ∈ {1, . . . , N… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Terms of (7) for n = 15, Fn = 610, Fn−1 = 377. The dashed lines are at y ∈  5 π4 , 5 16π4 , 5 25π4 , 5 81π4 , 5 121π4 [PITH_FULL_IMAGE:figures/full_fig_p007_1_3.png] view at source ↗
read the original abstract

We consider the asymptotics of sums of the form $$ \frac1{F_n^\sigma} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(\pi m/F_n)}\right|^\sigma} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(\pi F_{n-1}m/F_n)}\right|^\sigma} $$ where $(F_n)_{n \in \mathbb N} = (1, 1, 2, 3, 5, 8, 13, \dots)$ are the Fibonacci numbers. Such sums appear, for example, in the context of discrepancy theory and numerical integration methods reformulated as energy minimization problems. We show that for parameters $\sigma > 1$ and a large class of functions $f$ the above sum behaves asymptotically like $$ C n + D + O\left((1-\varepsilon)^{n}\right) $$ for some constants $C$ and $D$. These constants can be given via infinite series connected to the Dedekind zeta function over the algebraic number field $\mathbb Q(\sqrt5)$. In special cases we even observe simple closed-form expressions for such sums as above, explicitly proving that $$ \sum_{m=1}^{F_n-1} \frac1{\sin(\pi m/F_n)^2} \frac1{\sin(\pi F_{n-1} m/F_n)^2} = \frac{4n}{75} F_{2n} - \frac{17}{225}F_n^2 - (-1)^n \frac2{15} - \frac19. $$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper examines the asymptotic behavior of the normalized sum (1/F_n^σ) ∑_{m=1}^{F_n-1} [f(m/F_n)/|sin(π m/F_n)|^σ] [f(F_{n-1} m/F_n)/|sin(π F_{n-1} m/F_n)|^σ] for Fibonacci numbers F_n, σ > 1, and a class of functions f. It claims this sum equals C n + D + O((1-ε)^n) for constants C, D given by convergent series tied to the Dedekind zeta function of Q(√5). For the special case σ=2 and f≡1, it proves the exact identity ∑_{m=1}^{F_n-1} 1/sin(π m/F_n)^2 * 1/sin(π F_{n-1} m/F_n)^2 = (4n/75) F_{2n} - (17/225) F_n^2 - (-1)^n (2/15) - 1/9.

Significance. If the derivations hold, the work supplies precise asymptotics with exponential error for tensor-product energies on Fibonacci lattices, directly linking the leading coefficients to the Dedekind zeta function of Q(√5) via independent series. The explicit closed-form identity for the special case is a clear strength, as both sides are elementary and verifiable for each n, supporting applications in discrepancy theory and energy minimization.

major comments (2)
  1. [Main asymptotic result] Main asymptotic theorem (statement following the abstract and developed in the body): the decomposition of the sum via the Fibonacci recurrence and the ring of integers of Q(√5) is load-bearing for both the linear term C n and the exponential error; the manuscript must explicitly identify the precise regularity class on f (e.g., C^1 or Fourier-coefficient decay) that guarantees the error bound is uniform in that class.
  2. [Exact formula for σ=2, f=1] Special-case identity (displayed in the abstract and proved in the relevant section): while the closed form is elementary and checkable, the proof should include a direct verification step for small n (e.g., n=3,4) to confirm the coefficients 4/75, 17/225, etc., before invoking the general recurrence argument.
minor comments (2)
  1. [Abstract] The abstract refers to 'a large class of functions f' without definition; move the precise statement of the regularity hypothesis to the introduction or the statement of the main theorem.
  2. [Definition of C and D] Notation for the constants C and D: explicitly label the infinite series expressions in terms of the Dedekind zeta function so that readers can see they are independent of the original sum.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and recommendation of minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Main asymptotic result] Main asymptotic theorem (statement following the abstract and developed in the body): the decomposition of the sum via the Fibonacci recurrence and the ring of integers of Q(√5) is load-bearing for both the linear term C n and the exponential error; the manuscript must explicitly identify the precise regularity class on f (e.g., C^1 or Fourier-coefficient decay) that guarantees the error bound is uniform in that class.

    Authors: We agree that an explicit regularity class on f is necessary to make the uniformity of the O((1-ε)^n) error bound fully rigorous. The current manuscript refers to 'a large class of functions f' without a precise definition. In the revised version we will add the following statement to the main theorem: f is C^1 on the circle with ||f'||_∞ bounded and Fourier coefficients satisfying |ˆf(k)| ≤ C/|k|^2. This class is sufficient for the Fibonacci-recurrence decomposition and the estimates that connect the leading coefficients to the Dedekind zeta function of Q(√5). revision: yes

  2. Referee: [Exact formula for σ=2, f=1] Special-case identity (displayed in the abstract and proved in the relevant section): while the closed form is elementary and checkable, the proof should include a direct verification step for small n (e.g., n=3,4) to confirm the coefficients 4/75, 17/225, etc., before invoking the general recurrence argument.

    Authors: We accept the suggestion. The proof currently proceeds directly from the recurrence relation satisfied by the sums. In the revised manuscript we will insert an explicit verification subsection that computes both sides of the identity for n=3 and n=4 by direct summation (using the known values of F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21) and confirms that the numerical values match the closed-form expression with the stated coefficients 4/75, 17/225, etc. This check precedes the general recurrence argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation decomposes the normalized double sum using the linear recurrence of the Fibonacci sequence and the algebraic structure of the ring of integers in Q(√5). Leading coefficients C and D are expressed as independent convergent series attached to the Dedekind zeta function of that field; these series are not defined in terms of the target sum. The exponential error bound follows from the contraction |φ^{-2}| < 1. The special-case closed-form identity is an explicit polynomial expression in n, F_n and F_{2n} that is asserted to hold identically for integer n ≥ 2 and is therefore directly verifiable without reference to the asymptotic analysis. No load-bearing step reduces a claimed result to a fitted parameter, self-citation, or input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard analytic number theory applied to the Fibonacci recurrence and the algebraic structure of Q(√5); no new free parameters or postulated entities are introduced.

axioms (2)
  • standard math The Fibonacci sequence satisfies the recurrence F_n = F_{n-1} + F_{n-2} with initial conditions F_1=1, F_2=1.
    This recurrence defines the lattice indices and the relations used to analyze the sums.
  • standard math Analytic continuation and functional equation properties of the Dedekind zeta function over the quadratic field Q(√5).
    These properties are invoked to express the constants C and D as convergent series.

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    Feng-Zhen Zhao and Tianming Wang. Some results on generalized Fibonacci and Lucas numbers and Dedekind sums.Fibonacci Quart., 42(3):250–255, 2004. Appendix A. Auxiliary results A.1.Floor function Here we collect some results concerning the floor function that are used throughout Section 2. Similar relations have been proven and used for example in [18, 34...

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    (ii):Ifx=y, both sides are 0 and the inequality is obviously true

    The statement now follows by taking the inverses. (ii):Ifx=y, both sides are 0 and the inequality is obviously true. Therefore, without loss of generality letx < yand consider the ratio R(x, y) := 1 sin(πx)σ − 1 sin(πy) σ 1 x + 1 y σ−1 1 x − 1 y = xσyσ|sin(πy) σ −sin(πx) σ| sin(πx)σ sin(πy) σ(x+y) σ−1|y−x| . By (i),x σ sin(πx)−σ ≤1, respectively fory. Fur...

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