论文标题
地球跳
Globe-hopping
论文作者
论文摘要
我们考虑了与贝尔不平等相关的圆圈和球体上蚱hopper问题的版本（Goulko和Kent，2017年）。对于一个圆周$2π$的圆圈，我们表明，对于任何长度和任意跳跃长度的不受约束的草坪，蚱hopper的跳跃可能留在草坪上的概率的至高无上是一个。对于反植物草坪，根据定义，该草坪完全包含每对相反点中的一个并且具有长度$π$，我们证明这是正确的，除非跳高长度$ ϕ $是$ $π\ frac {p} {p} {p} {p，q $ p，q $ coprime和$ p $奇数。对于这些跳高长度，我们表明最佳概率为$ 1-1/Q $，构造最佳草坪。对于一对抗焦点草坪，我们表明，从一个跳到另一个的$ 1-1/Q $对于$ P，Q $ coprime，$ p $ odd和$ q $偶，以及一个在所有其他情况下。对于球体上的抗虫草坪，众所周知（Kent和Pitalúa-García，2014年），如果$ ϕ =π/q $，其中$ q \ in \ mathbb n $，则为grasshopper跳高的最佳保留概率$ 1-1/q $，是由半hemissphericallical lawn提供的。我们表明，在所有其他情况下，$ 0 <ϕ <π/2 $，半球形草坪不是最佳的，可以反驳半球形的最大值假设（Kent和Pitalúa-García，2014年）。我们讨论对贝尔实验和相关加密测试的影响。
We consider versions of the grasshopper problem (Goulko and Kent, 2017) on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference $2π$, we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper's jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length $π$, we show this is true except when the jump length $ϕ$ is of the form $π\frac{p}{q}$ with $p,q$ coprime and $p$ odd. For these jump lengths we show the optimal probability is $1 - 1/q$ and construct optimal lawns. For a pair of antipodal lawns, we show that the optimal probability of jumping from one onto the other is $1 - 1/q$ for $p,q$ coprime, $p$ odd and $q$ even, and one in all other cases. For an antipodal lawn on the sphere, it is known (Kent and Pitalúa-García, 2014) that if $ϕ= π/q$, where $q \in \mathbb N$, then the optimal retention probability of $1-1/q$ for the grasshopper's jump is provided by a hemispherical lawn. We show that in all other cases where $0<ϕ< π/2$, hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses (Kent and Pitalúa-García, 2014). We discuss the implications for Bell experiments and related cryptographic tests.
