[Paper Review] Contribution to Vojtech Jarnik

Nikolay Moshchevitin|arXiv (Cornell University)|Dec 12, 2009

Language and Culture2 references3 citations

TL;DR

This paper improves Vojtěch Jarník's classical theorems on Diophantine approximation exponents by establishing sharper lower bounds for individual Diophantine exponents β(Θ) in three key cases: m=1,n=3; m=3,n=1; and m=n=2. Using geometric and lattice-theoretic methods, including Minkowski's convex body theorem and analysis of best approximations, the authors derive new explicit functions g₁(α), g₂(α), and g₃(α) such that β(Θ) ≥ α(Θ)·g(α(Θ)), which strictly improve upon Jarník’s original inequalities in specified ranges of the uniform exponent α(Θ).

ABSTRACT

We find new inequalities between uniform and individual Diophantine exponents for three-dimensional Diophantine approximations. Also we give a result for two linear forms in two variables. The results improves V.Jarnik's theorem (1954).

Motivation & Objective

• To refine Vojtěch Jarník’s 1954 theorems on uniform and individual Diophantine exponents in multi-dimensional approximation.
• To address the gap in existing bounds for β(Θ) when m=1,n=3; m=3,n=1; and m=n=2, where Jarník’s inequalities are suboptimal.
• To establish new, improved lower bounds for the individual Diophantine exponent β(Θ) in terms of the uniform exponent α(Θ), using geometric and analytic techniques.
• To demonstrate that the new bounds are strictly better than Jarník’s in specific ranges of α(Θ), particularly for α(Θ) ∈ (1/3, α₀) in the m=1,n=3 case and α(Θ) > 1 in the m=n=2 case.
• To provide a sketched proof framework based on best approximations and Minkowski’s convex body theorem, applicable to linear forms in multiple variables.

Proposed method

• Define the uniform Diophantine exponent α(Θ) as the supremum of γ for which t^γ ψ_Θ(t) remains bounded as t→∞.
• Define the individual Diophantine exponent β(Θ) as the supremum of γ for which t^γ ψ_Θ(t) has finite liminf.
• Introduce new functions g₁(α), g₂(α), and g₃(α) as solutions to specific quadratic or cubic equations derived from geometric constraints.
• Use the theory of best approximations (x_ν) and associated vectors z_ν ∈ ℤ^{m+n} to analyze the decay rate of ||L_j(x)||.
• Apply Minkowski’s convex body theorem to two-dimensional lattices Λ ⊂ ℤ^{d} (d = m+n) to bound the product ζ_ν M_{ν+1} via the lattice determinant and angle δ between subspaces.
• Derive asymptotic inequalities involving ψ(t) = t^{-α+ε} and use the growth rates of M_ν to compare ψ(M_ν) with ζ_ν, leading to improved bounds on β(Θ).

Experimental results

Research questions

• RQ1 Can Jarník’s 1954 inequalities for Diophantine exponents be improved in the cases m=1,n=3, m=3,n=1, and m=n=2?
• RQ2 What is the optimal lower bound for the individual Diophantine exponent β(Θ) in terms of the uniform exponent α(Θ) for these three cases?
• RQ3 Are the new bounds strictly better than Jarník’s original inequalities in any range of α(Θ)?
• RQ4 How do the geometric structure of best approximations and lattice subspaces influence the decay rate of ||L_j(x)||?
• RQ5 Can the method of best approximations and Minkowski’s theorem be extended to derive tighter bounds than previously known?

Key findings

• For m=1, n=3 with three numbers linearly independent over ℤ together with 1, the bound β(Θ) ≥ α(Θ)g₁(α(Θ)) improves Jarník’s inequality in the range 1/3 < α(Θ) < α₀, where α₀ ≈ 0.545 is the real root of x³ - x² + 2x - 1 = 0.
• For m=3, n=1 with three numbers linearly independent over ℤ together with 1, the bound β(Θ) ≥ α(Θ)g₂(α(Θ)) improves Jarník’s inequality (3) for all α(Θ) ≥ 3, with g₂(α) = √(α + 1/α² - 7/4) + 1/α - 1/2.
• For m=n=2 with four numbers linearly independent over ℤ together with 1, the bound β(Θ) ≥ α(Θ)g₃(α(Θ)) improves Jarník’s inequality (2) in the range 1 < α(Θ) < ((1+√5)/2)² ≈ 2.618, where g₃(α) solves αx² + (α-1)x - (2α² - 2α + 1) = 0.
• The new bounds are strictly better than Jarník’s in their respective ranges, as shown by the fact that g₁(α) > (α²)/(1−α) for α ∈ (1/3, α₀), and similarly for g₂ and g₃.
• The proof relies on analyzing sequences of best approximations (x_ν) and their associated vectors z_ν ∈ ℤ^{m+n}, using Minkowski’s theorem on the lattice Λ = π ∩ ℤ^{d} for a 2D subspace π.
• The asymptotic analysis of the growth of M_ν and the decay of ψ(t) = t^{-α+ε} leads to the conclusion that ζ_ν ≪ ψ(M_ν^{g(α)/h(α)}) for appropriate g and h, which implies the improved bound on β(Θ).