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The distribution of the lattice points on circles. (English) Zbl 0777.11036
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.
Reviewer: W.G.Nowak (Wien)

##### MSC:
 11P21 Lattice points in specified regions 11L05 Gauss and Kloosterman sums; generalizations
##### Keywords:
distribution of lattice points on circles
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