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#### Ramanujan graphs and exponential sums over function fields

##### External Resource

https://doi.org/10.1016/j.jnt.2020.05.010

(Publisher version)

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##### Fulltext (public)

1909.07365.pdf

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##### Citation

Sardari, N. T., & Zargar, M. (2020). Ramanujan graphs and exponential sums over
function fields.* Journal of Number Theory,* *217*, 44-77.
doi:10.1016/j.jnt.2020.05.010.

Cite as: https://hdl.handle.net/21.11116/0000-0007-2D73-6

##### 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.

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.