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

Abstract

We consider the asymptotics of sums of the form $$\frac{1}{F_n^{\sigma }}\sum _{m=1}^{F_n-1}\frac{f(m/F_n)}{{\left|\sin (\pi m/F_n)\right|}^{\sigma }}\frac{f(F_{n-1}m/F_n)}{{\left|\sin (\pi F_{n-1}m/F_n)\right|}^{\sigma }}$$ 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 $$Cn+D+O\left((1-\epsilon {)}^n\right)$$ 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 $\mathbb{Q}(\sqrt{5})$. In special cases we even observe simple closed-form expressions for such sums as above, explicitly proving that $$\sum _{m=1}^{F_n-1}\frac{1}{\sin (\pi m/F_n{)}^2}\frac{1}{\sin (\pi F_{n-1}m/F_n{)}^2}=\frac{4n}{75}{F}_{2n}-\frac{17}{225}{F}_n^2-(-1{)}^n\frac{2}{15}-\frac{1}{9}.$$