PreprintCC BY 4.0arXiv
Error terms for sums of two linear transformations
Abstract
Let d be a positive integer, and let ℒ1,ℒ2 be irreducible and coprime linear transformations from ℤd to ℤd. In this paper, we prove that, for every finite nonempty set A⊆ℤd, |ℒ1A+ℒ2A| ≥ (|detℒ1|1/d+|detℒ2|1/d)d|A|−O(|A|1−1/d). This confirms the conjecture of Conlon and Lim on the error term for two linear transformations.
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BibTeX
@misc{yan2026error,
title = {{Error terms for sums of two linear transformations}},
author = {Xiao-Hui Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39235},
eprint = {2610.39235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39235},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Error terms for sums of two linear transformations
AU - Xiao-Hui Yan
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39235
UR - https://arxiv.org/abs/2610.39235
AB - Let d be a positive integer, and let ℒ_1,ℒ_2 be irreducible and coprime linear transformations from ℤ^d to ℤ^d. In this paper, we prove that, for every finite nonempty set A⊆ℤ^d, |ℒ_1A+ℒ_2A| ≥ (|detℒ_1|^1/d+|detℒ_2|^1/d)^d|A|−O(|A|^1−1/d). This confirms the conjecture of Conlon and Lim on the error term for two linear transformations.
ER -