On cellular rational approximations to $\hat{I}\P (5)$

Abstract

We analyse a certain family of cellular integrals, which are period integrals on the moduli space ${\mathcal{M}}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $\hat{I}\P (3)$ and $\hat{I}\P (5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p/q$ to $\hat{I}\P (5)$ satisfying \[ 0<\bigg|ζ(5)-\frac pq\bigg|<\frac1{q^{0.86}}. \]

32 pages, 2 figures