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Let $\mathscr{M}$ be a compact submanifold of $\mathbb{R}^{M}$. In thisarticle we establish an asymptotic formula for the number of rational pointswithin a given distance to $\mathscr{M}$ and with bounded denominators underthe assumption that $\mathscr{M}$ fulfills a certain curvature condition. Ourresult generalizes earlier work from Schindler and Yamagishi, as our curvaturecondition is a relaxation of that used by them. We are able to recover asimilar result concerning a conjecture by Huang and a slightly weaker analogueof Serre's dimension growth conjecture for compact submanifolds of$\mathbb{R}^{M}$.

On the number of rational points close to a compact manifold under a less restrictive curvature condition