Buffon's needle problem and related problems considered in the theory of uniform distribution (Q1185854)

The well-known needle-problem of Buffon is considered. An upper bound is given for the failure in the empirical estimation of \(2/\pi\). This bound depends on the discrepancy \(D_ N\) of the data \((\varphi_ 1,\eta_ 1),\ldots,(\varphi_ N,\eta_ N)\) with \(0\leq\varphi_ j\), \(\eta_ j\leq 1\), where \(D_ N=\sup_ J| N(J)/N-F(J)|\), the supremum is taken over all rectangles \(J\) in \([0,1]^ 2\), \(F(J)\) denotes area of \(J\), and \(N(J)=\#\{i:(\varphi_ i,\eta_ i)\in J\}\). As a consequence of a much more general bound, one gets in particular that \(| A/N- 2/\pi|\leq 40\sqrt{D_ N}\), where \(A=\{i:(\varphi_ i,\eta_ i)\in J\}\). As a consequence of a much more general bound, one gets in particular that \(| A/N-2/\pi|\leq 40\sqrt{D_ N}\), where \(A=\#\{i:\eta_ i\leq\cos\varphi_ i\}\). Analogous bounds are derived when the \(\varphi_ i,\eta_ i\) do not belong to the unit interval, but to a more general (multidimensional) set like the unit cube or the unit sphere.