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

TL;DR

This paper derives precise asymptotic formulas for sums involving Fibonacci numbers and sine functions, revealing their growth behavior and connections to algebraic number theory, with explicit formulas in special cases.

Contribution

It provides exact asymptotics and formulas for tensor product energies of Fibonacci lattices, linking them to Dedekind zeta functions and revealing closed-form expressions.

Findings

Sum behaves asymptotically like C n + D with exponential error term.

Constants C and D are expressed via infinite series related to Dedekind zeta functions.

Explicit closed-form for a special sum involving Fibonacci numbers and sine functions.

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 $$ \sum_{m=1}^{F_n-1} \frac1{\sin(\pi m/F_n)^2} \frac1{\sin(\pi F_{n-1} m/F_n)^2} = \frac{4n}{75} F_{2n} - \frac{17}{225}F_n^2 - (-1)^n \frac2{15} - \frac19. $$