On a problem connected with a theorem of Jarnik

Abstract

Divide the plane into unit squares and plot a rectifiable closed jordan curve of lengthl. We show that the error created when approximating the number of unit squares contained inside this curve by the area enclosed by it is less than αl, where α is a constant satisfying $$\frac{{4 + \pi }}{{2\pi }} \leqslant \alpha \leqslant 3 + \frac{2}{{6\pi (1 + \sqrt {1 + 2/9\pi } )}}.$$