PreprintCC BY 4.0arXiv
Sharp Bounds for Toral Eigenfunctions and Threefold Additive Energy on Circles
Abstract
For the lattice circle Pn={z∈ℤ2:|z|2=n}, we prove that the threefold additive energy J3(Pn)≤ C|Pn|3, with an absolute constant C. This establishes the optimal bound suggested by Krishnapur--Kurlberg--Wigman and Bombieri--Bourgain. We also prove a uniform weak L6 bound for L2-normalized eigenfunctions on the square torus, and hence uniform Lp bounds for every 1≤ p<6. This extends the classical L4 bounds of Cooke and Zygmund and establishes Bourgain's conjecture in this range. Applications include mean-square convergence for L6/5 observables and a logarithm-free remainder in the nodal length variance of arithmetic random waves.
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BibTeX
@misc{xie2026sharp,
title = {{Sharp Bounds for Toral Eigenfunctions and Threefold Additive Energy on Circles}},
author = {Guiyu Xie and Cheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37423},
eprint = {2610.37423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37423},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Bounds for Toral Eigenfunctions and Threefold Additive Energy on Circles
AU - Guiyu Xie
AU - Cheng Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37423
UR - https://arxiv.org/abs/2610.37423
AB - For the lattice circle P_n={z∈ℤ^2:|z|^2=n}, we prove that the threefold additive energy J_3(P_n)≤ C|P_n|^3, with an absolute constant C. This establishes the optimal bound suggested by Krishnapur--Kurlberg--Wigman and Bombieri--Bourgain. We also prove a uniform weak L^6 bound for L^2-normalized eigenfunctions on the square torus, and hence uniform L^p bounds for every 1≤ p<6. This extends the classical L^4 bounds of Cooke and Zygmund and establishes Bourgain's conjecture in this range. Applications include mean-square convergence for L^6/5 observables and a logarithm-free remainder in the nodal length variance of arithmetic random waves.
ER -