Let Γ ⊂ ℝ^d be a bounded strictly convex surface. We prove that the number k_n(Γ) of points of Γ that lie on the lattice 1/n ℤ² satisfies the following estimates: lim inf k_n(Γ)/n^d-2 < ∞ for d < 3 and lim inf k_n(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles.