PreprintCC BY 4.0arXiv
Gaps theorems for linear forms with cubic coefficients
Abstract
Let 1,α1,α2 be a basis for a cubic number field K⊂ℝ, and for τ1,τ2≥ 1 let G(α,τ) denote the number of distinct gaps between the fractional parts of the numbers m1α1+m2α2, with m1,m2∈ℤ, 0≤ m1<τ1 and 0≤ m2<τ2. We show that G(α,τ) is unbounded as τ varies. When K is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When K has a complex embedding we give an explicit construction, which also shows that there are constants c,κ>0 for which G(α,τ)≥ c(logmax{τ1,τ2})κ for infinitely many τ∈ℕ2.
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BibTeX
@misc{haynes2026gaps,
title = {{Gaps theorems for linear forms with cubic coefficients}},
author = {Alan Haynes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07471},
eprint = {2610.07471},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07471},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaps theorems for linear forms with cubic coefficients
AU - Alan Haynes
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07471
UR - https://arxiv.org/abs/2610.07471
AB - Let 1,α_1,α_2 be a basis for a cubic number field K⊂ℝ, and for τ_1,τ_2≥ 1 let G(α,τ) denote the number of distinct gaps between the fractional parts of the numbers m_1α_1+m_2α_2, with m_1,m_2∈ℤ, 0≤ m_1<τ_1 and 0≤ m_2<τ_2. We show that G(α,τ) is unbounded as τ varies. When K is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When K has a complex embedding we give an explicit construction, which also shows that there are constants c,κ>0 for which G(α,τ)≥ c(logmax{τ_1,τ_2})^κ for infinitely many τ∈ℕ^2.
ER -