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Globe-hopping.

Dmitry Chistikov1, Olga Goulko2, Adrian Kent3,4

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Summary
This summary is machine-generated.

This study analyzes the grasshopper problem on circles and spheres, relevant to Bell inequalities. Optimal probabilities for grasshoppers staying on lawns or jumping between them are determined, with implications for Bell experiments.

Keywords:
Bell inequalitiesgeometric combinatoricsgrasshopper problem

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Area of Science:

  • Mathematical Physics
  • Quantum Information Theory
  • Probability Theory

Background:

  • The grasshopper problem, previously studied on circles and spheres, is relevant to Bell inequalities and quantum cryptography.
  • Existing research established optimal probabilities for specific grasshopper jump lengths and lawn configurations on spheres.

Purpose of the Study:

  • To analyze the grasshopper problem on circles and spheres with arbitrary jump lengths and unconstrained/antipodal lawns.
  • To determine the optimal probabilities for a grasshopper staying on a lawn and jumping between antipodal lawns.
  • To investigate the optimality of hemispherical lawns on spheres and its implications for Bell experiments.

Main Methods:

  • Mathematical analysis of probability calculations for grasshopper jumps on circular and spherical geometries.
  • Construction of optimal lawns for specific jump lengths to achieve maximum probabilities.
  • Comparison of probabilities obtained with hemispherical lawns against theoretically derived optimal probabilities.

Main Results:

  • For unconstrained lawns on a circle, the supremum probability of staying on the lawn is one.
  • For antipodal lawns on a circle, the optimal probability is 1-1/q for specific jump lengths (πp/q, p odd), and one otherwise.
  • Hemispherical lawns are not always optimal on spheres for antipodal lawns, disproving previous hypotheses.

Conclusions:

  • The study provides precise optimal probabilities for the grasshopper problem on circles and spheres under various conditions.
  • Results refine understanding of geometric probability in contexts relevant to Bell inequalities and quantum cryptography.
  • The findings challenge existing assumptions about optimal lawn configurations in spherical settings.