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Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors

Yuval Yifrach

TL;DR

This work addresses the problem of understanding p-adic Diophantine approximation for algebraic vectors by translating approximation properties into homogeneous dynamics on the space of unimodular lattices, X_n. Using the Dani correspondence, Hecke operators, and the structure of compact A-orbits, the authors provide a new dynamical proof that joint algebraic vectors satisfy the $p$-adic Littlewood conjecture in the form $\liminf_{k\to\infty} (k|k|_p)^{1/n} \| k\overline{\alpha} \|_\infty = 0$. They further develop a directional refinement, proving that the limiting distributions of displacement directions from $\varepsilon$-approximations are explicitly describable as pushforwards of algebraic measures on $X_n$ to the sphere, via a weighted min-vec construction. The results connect arithmetic properties of number fields to equidistribution phenomena for lattice orbits, offering a precise, measure-theoretic description of how directions of best approximations distribute, and extending previous work on vector Littlewood-type problems to a dynamical setting. Overall, the paper advances both the quantitative and qualitative understanding of p-adic approximation for algebraic vectors with potential implications for future rigidity and equidistribution questions in higher-rank homogeneous dynamics.

Abstract

For every vector $\overline α\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $qα-\overline p$. We focus on algebraic vectors, namely $\overline α=(α_1,\dots,α_n)$ such that $1, α_1, \dots, α_n$ span a rank $n$ number field. For these vectors, we investigate the size of their displacements as well as the distribution of their directions. We give a new proof to the result of Bugeaud in \cite{YannPAdic} saying that algebraic vectors $\overline α$ satisfy the $p$-adic Littlewood Conjecture. Namely, we prove that \begin{equation} \liminf_{k \to \infty} \left( k \abs{k}_p \right)^{1/n} \| k (α_1, \dots, α_n) \|_\infty = 0. \end{equation} Our new proof lets us classify all limiting distributions, with a special weighting, of the sequence of directions of the defects in the $\varepsilon$-approximations of $(α_1, \dots, α_n)$. Each such limiting measure is expressed as the pushforward of an algebraic measure on $X_n$ to the sphere.

Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors

TL;DR

This work addresses the problem of understanding p-adic Diophantine approximation for algebraic vectors by translating approximation properties into homogeneous dynamics on the space of unimodular lattices, X_n. Using the Dani correspondence, Hecke operators, and the structure of compact A-orbits, the authors provide a new dynamical proof that joint algebraic vectors satisfy the -adic Littlewood conjecture in the form . They further develop a directional refinement, proving that the limiting distributions of displacement directions from -approximations are explicitly describable as pushforwards of algebraic measures on to the sphere, via a weighted min-vec construction. The results connect arithmetic properties of number fields to equidistribution phenomena for lattice orbits, offering a precise, measure-theoretic description of how directions of best approximations distribute, and extending previous work on vector Littlewood-type problems to a dynamical setting. Overall, the paper advances both the quantitative and qualitative understanding of p-adic approximation for algebraic vectors with potential implications for future rigidity and equidistribution questions in higher-rank homogeneous dynamics.

Abstract

For every vector and for every rational approximation we can associate the displacement vector . We focus on algebraic vectors, namely such that span a rank number field. For these vectors, we investigate the size of their displacements as well as the distribution of their directions. We give a new proof to the result of Bugeaud in \cite{YannPAdic} saying that algebraic vectors satisfy the -adic Littlewood Conjecture. Namely, we prove that \begin{equation} \liminf_{k \to \infty} \left( k \abs{k}_p \right)^{1/n} \| k (α_1, \dots, α_n) \|_\infty = 0. \end{equation} Our new proof lets us classify all limiting distributions, with a special weighting, of the sequence of directions of the defects in the -approximations of . Each such limiting measure is expressed as the pushforward of an algebraic measure on to the sphere.
Paper Structure (8 sections, 9 theorems, 75 equations)

This paper contains 8 sections, 9 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. $p$-Adic Littlewood conjecture for algebraic vectors
  3. Directional $p$-Adic Littlewood
  4. Notation and Preliminaries
  5. Compact Orbits
  6. Hecke Neighbors
  7. Periodic $A$-orbits vs Unipotent Orbits
  8. Proofs of the main theorems

Key Result

Theorem 1.4

Let $p$ be a prime. Let $K$ be a totally real number field of degree $n$, and let $M \leq K$ be a lattice (the $\mathbb{Z}$-span of a basis for $K$). Fix an ordering $\sigma_1,\dots,\sigma_n$ of the natural embeddings $K\hookrightarrow \mathbb{R}$, and denote where $\sigma=(\sigma_1,\dots,\sigma_n)$. Write $x_M=g\mathbb{Z}^n$ for some $g \in \operatorname{SL}_n(\mathbb{R})$. Define, for any $k \i

Theorems & Definitions (39)

  • Conjecture 1.3: Margulis
  • Theorem 1.4
  • Conjecture 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Definition 1.9
  • Theorem 1.10
  • Remark 1.11: Comparison to YannPAdic
  • Definition 1.12
  • Definition 1.13
  • ...and 29 more