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We obtain density theorems for cuspidal automorphic representations of GLₙ over Q which fail the generalized Ramanujan conjecture at some place. We depart from previous approaches based on Kuznetsov-type trace formulae, and instead rely on L-function techniques. This improves recent results of Blomer near the threshold of the pointwise bounds.
Lichtman et al. (Sat,) studied this question.