Ramanujan graphs and exponential sums over function fields (Q2197504)

The \(q+1\)-regular Morgenstern Ramanujan graphs \(X^{q,g}\) (depending on \(g\in \mathbb F_q[t]\)) have diameter at most \[(\frac{4}{3}+\varepsilon)\log_q|X^{q,g}|+O_\varepsilon\tag{1}\] (at least for odd \(q\) and irreducible \(g\)) provided that a twisted Linnik-Selberg conjecture over \(F_q(t)\) is true. It is a very remarkable result. 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 [\textit{A. Lubotzky} et al., Combinatorica 8, No. 3, 261--277 (1988; Zbl 0661.05035)] using the Ramanujan bound on Fourier coefficients of modular forms. They construct infinite families of Ramanujan graphs that prove that \(\frac{4}{3}\) cannot be improved.