Generic absence of finite blocking for interior points of Birkhoff billiards

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Let $x$ and $y$ be points in a billiard table $M=M(\sigma)$ in$\mathbb{\mathbb{R}}^{2}$ that is bounded by a curve $\sigma$. Weassume $\sigma\in\Sigma_{r}$ with $r\geq2$, where $\Sigma_{r}$is the set of simple closed $C^{r}$ curves in $\mathbb{R}^{2}$ withpositive curvature. A subset $B$ of $M\setminus\{x,y\}$ is calleda blocking set for the pair $(x,y)$ if every billiard pathin $M$ from $x$ to $y$ passes through a point in $B$. If a finiteblocking set exists, the pair $(x,y)$ is called secure in$M;$ if not, it is called insecure. We show that for $\sigma$in a dense $G_{\delta}$ subset of $\Sigma_{r}$ with the $C^{r}$topology, there exists a dense $G_{\delta}$ subset $\mathcal{\mathcal{R}=R}(\sigma)$of $M(\sigma)\times M(\sigma)$ such that $(x,y)$ is insecure in$M(\sigma)$ for each $(x,y)\in\mathcal{R}$. In this sense, for thegeneric Birkhoff billiard, the generic pair of interior points isinsecure. This is related to a theorem of S. Tabachnikov [24]that $(x,y)$ is insecure for all sufficiently close points$x$ and $y$ on a strictly convex arc on the boundary of a smoothtable.