## Abstract

We study the defect (or ‘signed area’) distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario (‘arithmetic random waves’). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for ‘most’ energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct ‘esoteric’ eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt’s subspace theorem.

Original language | English |
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Article number | 6651 |

Pages (from-to) | 6651-6684 |

Journal | Nonlinearity |

Volume | 34 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2021 |

### Bibliographical note

Funding Information:We are indebted to Ze?v Rudnick for many stimulating discussions, and his comments on an earlier version of this manuscript, in particular, pertaining to lemma 5.3 on Diophantine approximations. The research leading to these results has received funding from the European Research Council under the European Union?s Seventh Framework Programme (FP7/2007-2013), ERC grant agreement no. 335141 (IW and NY). PK was partially supported by the Swedish Research Council (2016-03701, 2020-04036).

Publisher Copyright:

© 2021 IOP Publishing Ltd & London Mathematical Society.

## Keywords

- Defect distribution
- Laplace eigenfunctions
- Signed measure
- Standard torus

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy (all)
- Applied Mathematics