PreprintCC BY 4.0arXiv
Quantitative linear independence for square roots
Abstract
We consider the problem of finding lower bounds for integer linear combinations of √(a1),…,√(aK), where a1,…,aK are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for K≥ 8 and nonzero integers m1,…,mK, |∑n≤ K mn√(an)| > e(2K−1−1)/K − 1/2 (maxn≤ K|mn|√(an)· √K)−(2K−1−1). This inequality improves the dependence on K in the classical product bound.
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BibTeX
@misc{aymone2026quantitative,
title = {{Quantitative linear independence for square roots}},
author = {Marco Aymone and Samuel Figueredo and Siddharth Iyer and Christian Táfula},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14161},
eprint = {2609.14161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14161},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantitative linear independence for square roots
AU - Marco Aymone
AU - Samuel Figueredo
AU - Siddharth Iyer
AU - Christian Táfula
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.14161
UR - https://arxiv.org/abs/2609.14161
AB - We consider the problem of finding lower bounds for integer linear combinations of √(a_1),…,√(a_K), where a_1,…,a_K are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for K≥ 8 and nonzero integers m_1,…,m_K, |∑_n≤ K m_n√(a_n)| > e^(2^K−1−1)/K − 1/2 (max_n≤ K|m_n|√(a_n)· √K)^−(2^K−1−1). This inequality improves the dependence on K in the classical product bound.
ER -