A counter-example to the theorem of Hiemer and Snurnikov

On the finite blocking property

A planar polygonal billiard $¶$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $¶$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to $A$ meets one of the $B_i$'s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov (see \cite{Mo}), we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces. We prove that the only Veech surfaces wit…