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The Fully Inhomogeneous p-Adic Littlewood Conjecture
Abstract
We prove that for almost every α∈ℝ, the \textit{fully inhomogeneous p-adic Littlewood Conjecture} holds; namely, ∀ δ∈ℝ, ∀ κ∈ℤp, liminf|q|→+∞ q ⟨ qα+δ⟩ |q+κ|p=0. Here, ⟨·⟩ denotes the distance to the nearest integer and |·|p denotes the p-adic norm. Moreover, we prove this conjecture for every quadratic irrational α. This gives an affirmative answer to the p-adic version of a question posed by Cassels in [Cas59].
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BibTeX
@misc{aka2026fully,
title = {{The Fully Inhomogeneous $p$-Adic Littlewood Conjecture}},
author = {Menny Aka and Alexander Gorodnik and Pankaj Vishe and Yuval Yifrach},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26137},
eprint = {2609.26137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26137},
}
RIS
TY - UNPB
TI - The Fully Inhomogeneous p-Adic Littlewood Conjecture
AU - Menny Aka
AU - Alexander Gorodnik
AU - Pankaj Vishe
AU - Yuval Yifrach
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.26137
UR - https://arxiv.org/abs/2609.26137
AB - We prove that for almost every α∈ℝ, the \textit{fully inhomogeneous p-adic Littlewood Conjecture} holds; namely, ∀ δ∈ℝ, ∀ κ∈ℤ_p, liminf_|q|→+∞ q ⟨ qα+δ⟩ |q+κ|_p=0. Here, ⟨·⟩ denotes the distance to the nearest integer and |·|_p denotes the p-adic norm. Moreover, we prove this conjecture for every quadratic irrational α. This gives an affirmative answer to the p-adic version of a question posed by Cassels in [Cas59].
ER -