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Denote by $\Theta_r$ the set of such angles. I am inclined to believe that the set $\Theta_r$ satisfies a zero-one law, with a possible threshold $r_0$. (If $r< r_0$, then set $\Theta_r$ has measure zero, while if $r>r_0$ the complement of $\Theta_r$ has measure zero.)

Here is my "argument". Fix $r\in (0,\frac{1}{2})$. Observe that we have a map

$$T_r: S^1\to S^1$$

defined as follows. Shoot a ball touching the pin at the origin in the direction $\theta$. Denote by $P_r(\theta)\in\mathbb{Z}^2$ the location of the first pin touched by the ball. After it touches $P_r(\theta)$, the ball will continue traveling along a ray of angle $T_r(\theta)$.

Clearly $\theta\in \Theta_r \Rightarrow T_r(\theta)\in \Theta_r$. The map $T_r$ is bijective and the set $\Theta_r$ is $T$-invariant. I am inclined to believe that $T_r$ is ergodic with respect to a measure absolutely continuous with respect to the arclength measure on $S^1$. If this is the case then the zero-one phenomenon above holds. (Do not ask me why I have this ergodic belief.)

The threshold statement seems harder to "argue", but observe that if $r=0$ then no irrational angle belongs to $\Theta_r$

Oops! $T_r$ is indeed not injective and in fact it is the wrong map. Here is the correct map. First fatten the pins to disk of radii $r$, and reduce the ball to a point particle. Consider the cylinder $C=S^1\times [-\pi/2,\pi/2]$, where $S^1$ is the boundary of a fat pin. A particle leaves the boundary of this fat pin with a velocity making an angle $\phi\in [-\pi/2,\pi/2]$ with the outer normal to the boundary at the departure point.

Then $T_r: C\to C$. This is almost injective (problems do appear when the particle grazes the boundary of a pin.) However is measure preserving for a Liouville-type measure. associated to the geodesic flow on the plane with the fat pins removed. The ergodicity is not obvious, but the dynamics of maps such as $T_r$ are being investigated by ergodic theorists. (I'm not one.)

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Denote by $\Theta_r$ the set of such angles. I am inclined to believe that the set $\Theta_r$ satisfies a zero-one law, with a possible threshold $r_0$. (If $r< r_0$, then set $\Theta_r$ has measure zero, while if $r>r_0$ the complement of $\Theta_r$ has measure zero.)

Here is my "argument". Fix $r\in (0,\frac{1}{2})$. Observe that we have a map

$$T_r: S^1\to S^1$$

defined as follows. Shoot a ball touching the pin at the origin in the direction $\theta$. Denote by $P_r(\theta)\in\mathbb{Z}^2$ the location of the first pin touched by the ball. After it touches $P_r(\theta)$, the ball will continue traveling along a ray of angle $T_r(\theta)$.

Clearly $\theta\in \Theta_r \Rightarrow T_r(\theta)\in \Theta_r$. The map $T_r$ is bijective and the set $\Theta_r$ is $T$-invariant. I am inclined to believe that $T_r$ is ergodic with respect to a measure absolutely continuous with respect to the arclength measure on $S^1$. If this is the case then the zero-one phenomenon above holds. (Do not ask me why I have this ergodic belief.)

The threshold statement seems harder to "argue", but observe that if $r=0$ then no irrational angle belongs to $\Theta_r$