Estimates for the number of rational points on convex curves and surfaces

Let $\Gamma\subset \mathbb R^d$ be a bounded strictly convex surface. Denote by $k_n(\Gamma)$ the number of points in the set $\Gamma\cap\frac1n\mathbb Z^d$. We prove that $\liminf k_n(\Gamma)/n^{d-2}<\infty$ for $d\ge 3$ and $\liminf k_n(\Gamma)/\log n<\infty$ for $d=2$.