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PreprintCC BY-NC-ND 4.0arXiv

Extreme values and logarithm laws for toral translations

  • Sourav Das
  • Anish Ghosh
  • Sundara Narasimhan
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Abstract

We study extreme values of normalised r-th nearest neighbour distance functions in inhomogeneous Diophantine approximation, and prove Weibull and Fréchet limit laws for these functions when the shift satisfies a natural Diophantine condition. Using these extreme value laws we prove corresponding logarithm laws for these functions. Our approach is based on homogeneous dynamics. We adapt the Poisson approximation strategy of Björklund and Gorodnik (arXiv:2201.05116) combined with an effective multiple equidistribution result, which we deduce using results of Strömbergsson (arXiv:1309.6103) and of Björklund and Gorodnik (arXiv:2105.05468).

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Cite

BibTeXRIS
BibTeX
@misc{das2026extreme,
  title = {{Extreme values and logarithm laws for toral translations}},
  author = {Sourav Das and Anish Ghosh and Sundara Narasimhan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.23401},
  eprint = {2608.23401},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23401},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Extreme values and logarithm laws for toral translations
AU  - Sourav Das
AU  - Anish Ghosh
AU  - Sundara Narasimhan
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.23401
UR  - https://arxiv.org/abs/2608.23401
AB  - We study extreme values of normalised r-th nearest neighbour distance functions in inhomogeneous Diophantine approximation, and prove Weibull and Fréchet limit laws for these functions when the shift satisfies a natural Diophantine condition. Using these extreme value laws we prove corresponding logarithm laws for these functions. Our approach is based on homogeneous dynamics. We adapt the Poisson approximation strategy of Björklund and Gorodnik (arXiv:2201.05116) combined with an effective multiple equidistribution result, which we deduce using results of Strömbergsson (arXiv:1309.6103) and of Björklund and Gorodnik (arXiv:2105.05468).
ER  -