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In a series of papers due to Browning, Heath-Brown, and Salberger (some of which are joint work, some individual), they established that for any projective variety $X \subset \mathbb{P}^n$, they established that $$\displaystyle N(X;B) = O_{d, n, \epsilon} (B^{r + \epsilon}),$$ under suitable assumptions, where $N(X;B)$ denotes the set of rational points on $X$ with height at most $B \geq 1$, $d$ is the degree of $X$, and $r$ is the dimension of $X$. This was done by reducing to the case of hypersurfaces, since in the projective case it can be shown that a suitable projection always exists that maps $X$ to a hypersurface in $\mathbb{P}^{r+1}$. In the same papers they established that for an affine hypersurface $Y \subset \mathbb{A}^n$, the estimate $$M(Y; B) = O_{d,n, \epsilon}(B^{n-2+\epsilon}),$$ where $M(Y; B)$ denotes the set of integer points on $Y$ bounded by $B$.

However, the same estimate has not been established for arbitrary affine varieties over $\mathbb{A}^n$, most likely because the birational projection that is critical for reducing the arbitrary variety case to the hypersurface case is not available when counting integer points only, since integer points are not preserved under such a transformation. However, the estimate $$\displaystyle M(Y; B) = O_{d,n,\epsilon}(B^{r-1+\epsilon})$$ seem likely to hold. Has anyone been able to prove such a result, perhaps under more generous assumptions on the affine variety? Perhaps just the case when $Y$ is the intersection of two hypersurfaces of sufficiently large degree?

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For homogeneous varieties there is the overview article: MR1975458 (2004h:11031) Margulis, Gregory(1-YALE) Diophantine approximation, lattices and flows on homogeneous spaces. A panorama of number theory or the view from Baker's garden (Zürich, 1999), 280–310, Cambridge Univ. Press, Cambridge, 2002. –  Peter Michor Jun 12 '13 at 9:38
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