On a problem connected with a theorem of Jarnik (Q1181776)

Divide the plane into unit squares and plot a rectifiable closed Jordan curve of length \(\ell\). Let \(A\) be the area and \(M\) be the number of lattice points inside the curve. By a theorem of Jarnik, \(| A- M|<\ell\). Denoting by \(N\) the number of unit squares inside the curve, we show that \(A-N\leq\alpha\ell\) where \(\alpha\) is a positive constant satisfying \[ (4+\pi)/2\pi\leq\alpha\leq3+2/6\pi(1+\sqrt{1+2/9\pi}). \] To find the lower bound for \(\alpha\), we simply give an example of a curve for which \(A-N\leq({4+\pi \over 2\pi})\ell\). For the upper bound, we use the fact that any curve which goes through five unit squares must be of length at least 1, so that any closed curve which goes through \(k\) unit squares is of length at least \([k/3]\). Combining this with the fact that \(A-N<k\) we derive the given upper bound.