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Spacing statistics for a point scatterer on the cubic three-torus
Abstract
We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form s5Ψ(log2(s−2))+o(s5), where Ψ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.
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BibTeX
@misc{lutsko2026spacing,
title = {{Spacing statistics for a point scatterer on the cubic three-torus}},
author = {Christopher Lutsko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10031},
eprint = {2610.10031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10031},
}
RIS
TY - UNPB
TI - Spacing statistics for a point scatterer on the cubic three-torus
AU - Christopher Lutsko
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10031
UR - https://arxiv.org/abs/2610.10031
AB - We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form s^5Ψ(log_2(s^−2))+o(s^5), where Ψ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.
ER -