Academic Journal
Ramanujan graphs and exponential sums over function fields
| العنوان: | Ramanujan graphs and exponential sums over function fields |
|---|---|
| المؤلفون: | Sardari, N., Zargar, M. |
| المصدر: | Journal of Number Theory |
| سنة النشر: | 2020 |
| المجموعة: | Max Planck Society: MPG.PuRe |
| الوصف: | 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. |
| نوع الوثيقة: | article in journal/newspaper |
| وصف الملف: | application/pdf |
| اللغة: | English |
| Relation: | info:eu-repo/semantics/altIdentifier/arxiv/1909.07365; http://hdl.handle.net/21.11116/0000-0007-2D73-6; http://hdl.handle.net/21.11116/0000-0007-2D75-4 |
| الاتاحة: | http://hdl.handle.net/21.11116/0000-0007-2D73-6 http://hdl.handle.net/21.11116/0000-0007-2D75-4 |
| Rights: | info:eu-repo/semantics/openAccess ; http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
| رقم الانضمام: | edsbas.106ED343 |
| قاعدة البيانات: | BASE |
| الوصف غير متاح. |