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Let 
$\mathfrak {F}_n$ be the set of all cuspidal automorphic representations 
$\pi$ of 
$\mathrm {GL}_n$ with unitary central character over a number field 
$F$. We prove the first unconditional zero density estimate for the set 
$\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg 
$L$-functions, where 
$\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at 
$s=\frac {1}{2}$ for almost all 
$L(s,\pi \times \pi ')\in \mathcal {S}$; (ii) a strong on-average form of effective multiplicity one for almost all 
$\pi \in \mathfrak {F}_n$; and (iii) a positive level of distribution for 
$L(s,\pi \times \widetilde {\pi })$, in the sense of Bombieri–Vinogradov, for each 
$\pi \in \mathfrak {F}_n$.
