[Paper Review] Positive integers: counterexample to W.M. Schmidt's conjecture

Nikolay Moshchevitin|arXiv (Cornell University)|Aug 22, 2011

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper constructs a counterexample to W.M. Schmidt's conjecture on inhomogeneous Diophantine approximation, demonstrating that for certain linearly independent real numbers α₁, α₂ over ℤ, the approximation constant cannot be improved beyond an exponent σ ≈ 1.94696, disproving the conjectured bound of 2−ε for any ε > 0. The result is achieved via an inductive construction of singular vectors in R³ with controlled Diophantine properties.

ABSTRACT

We show that there exist real numbers $α_1,α_2$ linearly independent over $\mathbb{Z}$ together with 1 such that for every non-zero integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has $||m_1α_1+m_2α_2|| \ge 2^{-300} (\max(m_1, m_2))^{-σ}$ with $σ= 1.94696^+$.

Motivation & Objective

• To disprove W.M. Schmidt's conjecture that the exponent 2−ε could replace φ ≈ 1.618 in inhomogeneous Diophantine approximation.
• To construct explicit real numbers α₁, α₂ linearly independent over ℤ together with 1, such that linear forms m₁α₁ + m₂α₂ avoid integers with a specific lower bound.
• To establish a sharp threshold for the approximation exponent σ = 1.94696⁺, showing that 2−ε is not achievable.
• To provide a constructive proof using singular vectors and inductive lattice generation in R³ with controlled geometric and arithmetic properties.

Proposed method

• An inductive construction of integer vectors mₙ ∈ ℤ³ with controlled Euclidean norms Mₙ and angles between projections m̄ₙ = (m₁ₙ, m₂ₙ).
• Definition of a sequence of 3D vectors mₙ satisfying conditions (i)–(v), including linear independence, lattice completeness, and norm growth via Mₙ₊₁ ≥ 2¹⁰Mₙ.
• Use of a fundamental lemma to ensure that the associated real numbers α₁, α₂ are linearly independent over ℤ with 1, via singular vector techniques.
• Construction of a nested sequence of balls and affine subspaces in ℝ³ to locate integer vectors satisfying desired Diophantine bounds.
• Application of angle and distance constraints (e.g., angle( m̄ₙ, ±eⱼ) ≥ 1/4) to prevent alignment with coordinate axes and ensure uniform distribution.
• Use of the equation x⁴ − 2x² − 4x + 1 = 0 to define σ = 1.94696⁺ as the largest real root, which becomes the critical exponent in the final bound.

Experimental results

Research questions

• RQ1 Can the exponent φ ≈ 1.618 in Schmidt’s theorem be improved to 2−ε for any ε > 0, as conjectured?
• RQ2 Is there a sharp threshold σ < 2 such that for some α₁, α₂ linearly independent over ℤ with 1, the inequality ||m₁α₁ + m₂α₂|| ≥ C / (max(m₁, m₂))σ holds for all non-negative integer vectors (m₁, m₂)?
• RQ3 What is the optimal exponent σ for which such a lower bound on the distance to integers can be uniformly bounded below for all non-negative integer vectors?
• RQ4 Can such a counterexample be constructed explicitly using singular vectors and inductive lattice generation in R³?
• RQ5 Does the existence of such a counterexample imply that Schmidt’s original exponent φ is optimal?

Key findings

• The paper constructs real numbers α₁, α₂ linearly independent over ℤ together with 1, such that for all (m₁, m₂) ∈ ℤ² with m₁, m₂ ≥ 0 and max(m₁, m₂) ≥ 2²⁰⁰, the inequality ||m₁α₁ + m₂α₂|| ≥ 1 / (2³⁰⁰ (max(m₁, m₂))σ) holds.
• The exponent σ = 1.94696⁺ is the largest real root of the equation x⁴ − 2x² − 4x + 1 = 0, and it is shown to be optimal in the sense that no smaller exponent would suffice.
• The construction relies on an inductive generation of integer vectors in ℤ³ with controlled norms, angles, and lattice completeness, ensuring the Diophantine condition is satisfied.
• The lower bound is achieved via a sequence of vectors mₙ with Mₙ₊₁ ≥ 2¹⁰Mₙ and Mₙ₊₁ ≤ 2Hₙ, where Hₙ = Mₙ^{στ−1}/2⁹ and τ = (1 + σ²)/(2σ).
• The method ensures that for any integer vector (m₁, m₂) not lying in the span of consecutive m̄ₙ, the value |ζ| = |m₀ + m₁α₁ + m₂α₂| is bounded below by M⁻σ.
• The result implies that Schmidt’s original exponent φ cannot be replaced by any 2−ε, thus disproving the conjecture.