The distribution of the lattice points on circles (Q1208185)

This is one more enlightening contribution of the author concerning the distribution of lattice points on circles \(C_ n\): \(x^ 2+y^ 2=n\) where \(n\) is always such that \[ r(n)=\#\{(u,v)\in\mathbb{Z}^ 2:\;u^ 2+v^ 2=n\}\neq 0. \] As a measure of the uniformity of this distribution, the area \(S(n)\) of the polygon formed by all of the lattice points on \(C_ n\) is considered. It turns out (Theorem 1) that there exists an infinite sequence of integers \(n\) with \(r(n)\neq 0\) for which \[ S(n)=\pi n+O\left( n\left( {{\log\log n} \over {\log n}}\right)^ 2 \right). \] On the other hand, \(S(2^ k)\sim 2^{k+1}\). Actually, the author proves (Theorem 3) that the set \(\bigl\{S(n)/\pi n:\;r(n)\neq 0\bigr\}\) lies dense in the interval \([2/\pi,1]\). One more result (Theorem 2) shows that one can always pick \(n\) in such a way that \(r(n)\) becomes arbitrarily large and all the lattice points of \(C_ n\) lie in sectors close to the coordinate axes.