On Sarnak's Density Conjecture and its Applications

Abstract

Sarnak's Density Conjecture is an explicit bound on the multiplicities of nontempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue ([53]). The goal of this work is to discuss similar hypotheses, their interrelation and applications. We mainly focus on two properties -the spectral Spherical Density Hypothesis and the geometric Weak Injective Radius Property. Our results are strongest in the p-adic case, where we show that the two properties are equivalent, and both imply Sarnak's General Density Hypothesis. One possible application is that either the limit multiplicity property or the weak injective radius property imply Sarnak's Optimal Lifting Property ([52]). Conjecturally, all those properties should hold in great generality. We hope that this work will motivate their proofs in new cases.

Optimal Diophantine Exponents for $\mathrm{SL}(n)$

“…On the other hand, we do not require a strong bound of individual Hecke eigenvalue, but only the GRC on average. In general, one expects Sarnak's Density Hypothesis to hold; see [44,43,25,4,26,2] for various aspects of this hypothesis.…”

“…In our conditional result, we assume a certain form of the density hypothesis, namely Conjecture 1 which can be realized as a higher rank analogue of Sarnak's Density Hypothesis (see Proposition 4.12 and discussion there), and apply to our question. This approach was already used in different contexts to deduce results of a similar flavor (see [42,10,24,25]). The version that is relevant for us can be realized as Density relative to the Weyl's law.…”

The density conjecture for principal congruence subgroups

“…It is easy to see (cf. [GK,Section 2.6]) that a lower bound is given by T n(n−1) /q n 2 −1 + (T /q) n(n−1)/2 + 1.…”