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Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\theta$ w.r.t. the horizontal. It reflects from pins in the natural manner:
          PinBall http://cs.smith.edu/~orourke/MathOverflow/PinBall.jpg
What happens? More specifically,

Q. For a given $r$, what is the measure of the set of angles $\theta$ for which the pinball remains a finite distance from the origin forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to Polya's Orchard problem (e.g., explored in the MO question, "Efficient visibility blocker s in Polya’s orchard problem"), but the reflections produce complex interactions. It is also similar to Pach's enchanted forest problem, mentioned in the MO question, "Trapped rays bouncing between two convex bodies", but it seems simpler than that unsolved problem, due to the lattice regularity. Has it been considered before, in some guise? If so, pointers would be appreciated. Thanks!

Addendum. My question can be rephrased in terms of the "Sinai billiard," as Anthony Quas explains: I am asking for the fate of radial rays in the situation illustrated:
                          Sinai http://cs.smith.edu/~orourke/MathOverflow/Sinai.jpg

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\theta$ w.r.t. the horizontal. It reflects from pins in the natural manner:
         
What happens? More specifically,

Q. For a given $r$, what is the measure of the set of angles $\theta$ for which the pinball remains a finite distance from the origin forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to Polya's Orchard problem (e.g., explored in the MO question, "Efficient visibility blocker s in Polya’s orchard problem"), but the reflections produce complex interactions. It is also similar to Pach's enchanted forest problem, mentioned in the MO question, "Trapped rays bouncing between two convex bodies", but it seems simpler than that unsolved problem, due to the lattice regularity. Has it been considered before, in some guise? If so, pointers would be appreciated. Thanks!

Addendum. My question can be rephrased in terms of the "Sinai billiard," as Anthony Quas explains: I am asking for the fate of radial rays in the situation illustrated:
                         

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\theta$ w.r.t. the horizontal. It reflects from pins in the natural manner:
          PinBall http://cs.smith.edu/~orourke/MathOverflow/PinBall.jpg
What happens? More specifically,

Q. For a given $r$, what is the measure of the set of angles $\theta$ for which the pinball remains a finite distance from the origin forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to Polya's Orchard problem (e.g., explored in the MO question, "Efficient visibility blocker s in Polya’s orchard problem"), but the reflections produce complex interactions. It is also similar to Pach's enchanted forest problem, mentioned in the MO question, "Trapped rays bouncing between two convex bodies", but it seems simpler than that unsolved problem, due to the lattice regularity. Has it been considered before, in some guise? If so, pointers would be appreciated. Thanks!

Addendum. My question can be rephrased in terms of the "Sinai billiard," as Anthony Quas explains: I am asking for the fate of radial rays in the situation illustrated:
                          Sinai http://cs.smith.edu/~orourke/MathOverflow/Sinai.jpg

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\theta$ w.r.t. the horizontal. It reflects from pins in the natural manner:
          PinBall http://cs.smith.edu/~orourke/MathOverflow/PinBall.jpg
What happens? More specifically,

Q. For a given $r$, what is the measure of the set of angles $\theta$ for which the pinball remains a finite distance from the origin forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to Polya's Orchard problem (e.g., explored in the MO question, "Efficient visibility blocker s in Polya’s orchard problem"), but the reflections produce complex interactions. It is also similar to Pach's enchanted forest problem, mentioned in the MO question, "Trapped rays bouncing between two convex bodies", but it seems simpler than that unsolved problem, due to the lattice regularity. Has it been considered before, in some guise? If so, pointers would be appreciated. Thanks!

Addendum. My question can be rephrased in terms of the "Sinai billiard," as Anthony Quas explains: I am asking for the fate of radial rays in the situation illustrated:
                          Sinai http://cs.smith.edu/~orourke/MathOverflow/Sinai.jpg

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\theta$ w.r.t. the horizontal. It reflects from pins in the natural manner:
          PinBall http://cs.smith.edu/~orourke/MathOverflow/PinBall.jpg
What happens? More specifically,

Q. For a given $r$, what is the measure of the set of angles $\theta$ for which the pinball remains a finite distance from the origin forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to Polya's Orchard problem (e.g., explored in the MO question, "Efficient visibility blocker s in Polya’s orchard problem"), but the reflections produce complex interactions. It is also similar to Pach's enchanted forest problem, mentioned in the MO question, "Trapped rays bouncing between two convex bodies", but it seems simpler than that unsolved problem, due to the lattice regularity. Has it been considered before, in some guise? If so, pointers would be appreciated. Thanks!

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\theta$ w.r.t. the horizontal. It reflects from pins in the natural manner:
          PinBall http://cs.smith.edu/~orourke/MathOverflow/PinBall.jpg
What happens? More specifically,

Q. For a given $r$, what is the measure of the set of angles $\theta$ for which the pinball remains a finite distance from the origin forever?

Many other questions suggest themselves, but let me leave it at that basic question for now.

This problem seems superficially similar to Polya's Orchard problem (e.g., explored in the MO question, "Efficient visibility blocker s in Polya’s orchard problem"), but the reflections produce complex interactions. It is also similar to Pach's enchanted forest problem, mentioned in the MO question, "Trapped rays bouncing between two convex bodies", but it seems simpler than that unsolved problem, due to the lattice regularity. Has it been considered before, in some guise? If so, pointers would be appreciated. Thanks!

Addendum. My question can be rephrased in terms of the "Sinai billiard," as Anthony Quas explains: I am asking for the fate of radial rays in the situation illustrated:
                          Sinai http://cs.smith.edu/~orourke/MathOverflow/Sinai.jpg