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Normal and Lognormal Asymptotics for Lattice-Point Counts in Random Translates of High-Dimensional Balls

  • Xin Hang Ji
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Abstract

Let the center of a d-dimensional Euclidean ball be uniformly distributed modulo ℤd. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is 4π2Rd2/(d+2)=log d. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.

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Cite

BibTeXRIS
BibTeX
@misc{ji2026normal,
  title = {{Normal and Lognormal Asymptotics for Lattice-Point Counts in Random Translates of High-Dimensional Balls}},
  author = {Xin Hang Ji},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30810},
  eprint = {2608.30810},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30810},
}
RIS
TY  - UNPB
TI  - Normal and Lognormal Asymptotics for Lattice-Point Counts in Random Translates of High-Dimensional Balls
AU  - Xin Hang Ji
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30810
UR  - https://arxiv.org/abs/2608.30810
AB  - Let the center of a d-dimensional Euclidean ball be uniformly distributed modulo ℤ^d. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is 4π^2R_d^2/(d+2)=log d. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.
ER  -