Tropical formulae for summation over a part of SL(2, Z)

TL;DR

This paper investigates the convergence of a sum over lattice parallelograms related to triangle inequality defects, providing new convergence bounds and explicit formulas, with a novel tropical geometric approach.

Contribution

It introduces a method using tropical analogue of caustic curves to derive explicit summation formulas and improves convergence bounds for the sum over SL(2,Z) lattice parallelograms.

Findings

Proves convergence of F(s) for s>1 and divergence at s=1/2.

Derives an explicit sum formula equal to 1/24.

Introduces a tropical geometric method for obtaining such formulas.

Abstract

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$, let $(a,b,c,d)$ stand for $a,b,c,d\in\mathbb Z_{\geq 0}$ such that $ad-bc=1$. Define \begin{equation} \label{eq_main} F(s) = \sum_{(a,b,c,d)} f(a,b,c,d)^s. \end{equation} In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one. We prove that $F(s)$ converges when $s>1$ and diverges at $s=1/2$. (This papers differs from its published version: Fedor Petrov showed us how to easily prove that $F(s)$ converges for $s>2/3$ and diverges for $s\leq 2/3$, see below.) We also prove $$\sum\limits_{\substack{(a,b,c,d), 1\leq a\leq b, 1\leq c\leq d}} \frac{1}{(a+b)^2(c+d)^2(a+b+c+d)^2} = 1/24,$$ and show a general method to obtain such formulae. The method comes from the consideration of the tropical analogue of the caustic curves, whose moduli give a complete set of continuous invariants on the space of convex domains.