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Arithmetic purity of strong approximation for semi-simple simply connected groups

In this article we establish the arithmetic purity of strong approximation for certain semi-simple simply connected $k$-simple linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with codim$(G\setminus U, G)\geq 2$, we prove that (i) if $G$ is $k$-isotropic, then $U$ satisfies strong approximation off any one (hence any finitely many) place; (ii) if $G$ is the spin group of a non-degenerate quadratic form which is non-compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine combinatorial sieve which allows to produce integral points with almost prime polynomial values.