Let $\cal C$ be a circle in ${\mathbb R}^2$ : $\cal C=\lbrace (x,y)\in{\mathbb R}^2 | (x-x_0)^2+(y-y_0)^2=r^2\rbrace$ for some constants $x_0,y_0,r$.
What is the maximal number of points that can be contained in ${\cal C}\cap {\mathbb Z}^2$ ? I conjecture it is $4$, attained for the "trivial" case $x_0=y_0=0,r=1$.