## Abstract

We show that if A ⊂ [k]^{n}, then A is ε -close to a junta depending upon at most exp(O(|∂A|/(k^{n-1}ε))) coordinates, where ∂A denotes the edge-boundary of A in the l^{1} -grid. This bound is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [6], or as a characterisation of large subsets of the l^{1} -grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [1]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

Original language | English (US) |
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Pages (from-to) | 253-279 |

Number of pages | 27 |

Journal | Random Structures and Algorithms |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1 2016 |

### Bibliographical note

Publisher Copyright:© 2015 Wiley Periodicals, Inc.

## Keywords

- Boolean functions
- Lipschitz
- influence