Precise Asymptotics and Exact Formulas for Tensor Product Energies of Fibonacci Lattices
This provides rigorous mathematical foundations for discrepancy theory and numerical integration methods, but the results are incremental for the specific case of Fibonacci lattices.
The paper derives precise asymptotic formulas for tensor product energies of Fibonacci lattices, showing that sums over Fibonacci numbers behave like C n + D with explicit constants related to the Dedekind zeta function, and provides exact closed-form expressions in special cases.
We consider the asymptotics of sums of the form $$ \frac1{F_n^σ} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(πm/F_n)}\right|^σ} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(πF_{n-1}m/F_n)}\right|^σ} $$ 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 $σ> 1$ 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 $$ \sum_{m=1}^{F_n-1} \frac1{\sin(πm/F_n)^2} \frac1{\sin(πF_{n-1} m/F_n)^2} = \frac{4n}{75} F_{2n} - \frac{17}{225}F_n^2 - (-1)^n \frac2{15} - \frac19. $$