Ramanujan graphs and exponential sums over function fields

TL;DR

This paper establishes a new upper bound on the diameter of certain Ramanujan graphs over function fields, contingent on a conjecture, and provides unconditional constructions showing the bound's optimality.

Contribution

It proves a new diameter bound for Morgenstern Ramanujan graphs assuming a twisted Linnik-Selberg conjecture and constructs infinite families demonstrating the bound's sharpness.

Findings

New diameter bound of approximately (4/3) log_q |X^{q,g}| for Ramanujan graphs.

Conditional proof relies on a twisted Linnik-Selberg conjecture over function fields.

Unconditional constructions show the (4/3) factor cannot be improved.

Abstract

We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending on $g\in\mathbb{F}_q[t]$) have diameter at most $\left(\frac{4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$ (at least for odd $q$ and irreducible $g$) provided that a twisted Linnik-Selberg conjecture over $\mathbb{F}_q(t)$ is true. This would break the 30 year-old upper bound of $2\log_{q}|X^{q,g}|+O(1)$, a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We also unconditionally construct infinite families of Ramanujan graphs that prove that $\frac{4}{3}$ cannot be improved.