PreprintCC BY 4.0arXiv
The optimal scale in Linnik's microsquare problem for almost all integers
Abstract
We prove that, for any positive function Ψ(t)→∞, almost all positive integers n satisfying 4nmid n and nnot≡7pmod8 can be written as n=x2+y2+z2, where x,y,z∈ℤ and 1leqslant z<√(log n) Ψ(n). This improves Wooley's bound, and we show that the result is best possible. Our approach combines Hooley's weighted representation function for sums of two squares with Green's W-trick. We also give a necessary and sufficient condition, in terms of covering systems, for almost all odd integers to be represented as the sum of a squarefree number and a power of two, answering a question of Granville and Soundararajan.
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BibTeX
@misc{sun2026optimal,
title = {{The optimal scale in Linnik's microsquare problem for almost all integers}},
author = {Yu-Chen Sun and Lilu Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13894},
eprint = {2609.13894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The optimal scale in Linnik's microsquare problem for almost all integers
AU - Yu-Chen Sun
AU - Lilu Zhao
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13894
UR - https://arxiv.org/abs/2609.13894
AB - We prove that, for any positive function Ψ(t)→∞, almost all positive integers n satisfying 4nmid n and nnot≡7pmod8 can be written as n=x^2+y^2+z^2, where x,y,z∈ℤ and 1leqslant z<√(log n) Ψ(n). This improves Wooley's bound, and we show that the result is best possible. Our approach combines Hooley's weighted representation function for sums of two squares with Green's W-trick. We also give a necessary and sufficient condition, in terms of covering systems, for almost all odd integers to be represented as the sum of a squarefree number and a power of two, answering a question of Granville and Soundararajan.
ER -