A problem on limit superior Let $A_n$ be the square $[(x,y) : |x|\leq 1, |y|\leq 1]$ rotated through the angle $2\pi n\theta$. I need to find the geometric description of $lim sup_n A_n$ when $\theta$ irrational. I understand intuitively from the fact that the set $\{ frac(n \theta) : n \in \Bbb N\}$  ($frac(x)$ is fractional part of $x$) is dense in $[0,1]$ that lim sup will be the outer open circle of the unit square aligned with the axes. But how to formally write it i.e. given $k$ and a point $(p,q)$ how to find $n \gt k$ and $(x,y)$ in the original square (aligned with the $X$ and $Y$ axes) such that $(p,q)$ is the transformed version of $(x,y)$ in $A_n$
 A: First, any point at distance more than $\sqrt2$ from the origin is in $A_n$ for no $n$ hence it is not in $A=\limsup A_n$ either. 
Next, consider a point $(x,y)=(r\cos\alpha,r\sin\alpha)$  at distance less than $\sqrt2$ from the origin, thus $r\lt\sqrt2$. Let $C_\beta$ denote the square $[-1,1]^2$ rotated by the angle $\beta$. Then each $C_\beta$ contains the rays starting at $(0,0)$ of length $\sqrt2$ in the directions $\beta\pm\pi/4$ hence $(x,y)$ is in the interior of $C_{\alpha+\pi/4}$ and, by continuity, $(x,y)$ is in the interior of $C_\beta$ for every $\beta$ close enough to $\alpha+\pi/4$, say every $\beta$ in some interval $(\beta_1,\beta_2)\subset(0,2\pi)$ with positive length.
Since $\theta$ is irrational, there exists infinitely many integers $n$ such that the fractional part of $n\theta$ is in the interval $(\beta_1/2\pi,\beta_2/2\pi)\subset(0,1)$. For each such $n$, $C_{2\pi n\theta}=C_\beta$ for some $\beta$ in the interval $(\beta_1,\beta_2)$ hence $(x,y)$ is in $C_{2\pi n\theta}$. This happens for infinitely many $n$ hence $(x,y)$ is in $A$.
Finally, if $(x,y)$ is at distance exactly  $\sqrt2$ of the origin, say $(x,y)=(\sqrt2\cos\alpha,\sqrt2\sin\alpha)$, then $(x,y)$ is in $C_{2\pi n\theta}$ if and only if $2\pi n\theta=\alpha+\pi/4\pmod{\pi/2}$. This happens for at most one value of $n$ hence $(x,y)$ is not in $A$.
This proves that $A$ is the open disk centered at zero with radius $\sqrt2$.
