Bulletin of the Brazilian Mathematical Society, New Series ( IF 0.7 ) Pub Date : 2021-07-22 , DOI: 10.1007/s00574-021-00270-9 Mikhail Belolipetsky 1 , Matilde Lalín 2 , Plinio G P Murillo 3 , Lola Thompson 4
It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines \(c Q^{1/2} + O(Q^{1/4})\) square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to \(\frac{4}{3}Q^{3/2}+O(Q)\). Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3.
中文翻译:
计算算术双曲 3-Orbifolds 的 Salem 数
众所周知,算术双曲轨道的闭合测地线的长度与塞勒姆数有关。我们开始对这种现象进行定量研究。我们证明了任何非紧算术 3 维 orbifold 定义了\(c Q^{1/2} + O(Q^{1/4})\)小于或等于 4 次的可平方根 Salem 数到Q。这个数量可以与此类塞勒姆数的总数进行比较,这表明它是渐近于\(\frac{4}{3}Q^{3/2}+O(Q)\). 假设 Marklof 的间隙猜想,我们可以将这些结果扩展到紧凑的算术 3-orbifolds。作为一个应用程序,我们获得了非紧偶数维算术双折的测地线谱中平均多重性强指数增长的下界。以前,这种下限仅在维度 2 和 3 中获得。