A Uniform Version of Jarník's Theorem

Later on in [2], Grekos refined the previous results, in some cases, by introducing the infimum of the radii of curvature r(Γ ) of the curve. He succeeded in showing an upper bound of the shape N(Γ ) l(Γ )r(Γ )−1/3 and conversely constructed a family of curves Γ0 with N(Γ0) l(Γ0)r(Γ0)−1/3. Naturally, Grekos’ results suppose that the curve has at least C2-regularity. In fact, this is the maximal regularity one can have for a “uniform” bound as good as (1), because of Swinnerton-Dyer’s results in [8] showing that if Γ is C3, one has