[Paper Review] Nesterenko's linear independence criterion for vectors

Stéphane Fischler|arXiv (Cornell University)|Feb 10, 2012

Mathematical Dynamics and Fractals7 references4 citations

TL;DR

This paper generalizes Nesterenko's linear independence criterion from scalar values to vectors, establishing a lower bound on the rational rank of a family of $ p $ vectors in $ \mathbb{R}^k $ using sequences of integer-coefficient linear forms that are small at $ k $ distinct points. The method relies on Minkowski's theorem on convex bodies and extends the Ball-Rivoal irrationality results for zeta values at odd integers by providing a stronger, quantitative criterion for linear independence over $ \mathbb{Q} $.

ABSTRACT

In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small at $k$ points. This is a generalization to vectors of Nesterenko's linear independence criterion (which corresponds to $k=1$), used by Ball-Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski's theorem on convex bodies.

Motivation & Objective

• Address the challenge of proving linear independence over $ \mathbb{Q} $ for families of real numbers, particularly values of the Riemann zeta function at odd integers.
• Overcome the limitation of existing methods that fail to improve the constant in the asymptotic lower bound for the dimension of the $ \mathbb{Q} $-span of $ \zeta(3), \zeta(5), \ldots, \zeta(a) $.
• Develop a vector-valued generalization of Nesterenko's criterion to enable sharper rank estimates in Diophantine approximation.
• Provide a quantitative, effective version of linear independence criteria applicable to families of vectors in $ \mathbb{R}^p $.
• Enable new proofs of irrationality results by reducing the need for constructing $ p $ linearly independent small forms, instead relying on a single sequence of forms.

Proposed method

• Apply Minkowski's convex body theorem to derive a lower bound on the $ \mathbb{Q} $-rank of a family of $ p $ vectors in $ \mathbb{R}^k $, viewed as a $ \mathbb{Q} $-vector space.
• Use a sequence of integer-coefficient linear forms $ L_n $ on $ \mathbb{R}^p $ that are small at $ k $ distinct points $ e_1, \ldots, e_k \in \mathbb{R}^p $, with controlled growth in coefficients.
• Establish that if $ |L_n(e_j)| = Q_n^{-\tau_j + o(1)} $ and $ \max |\ell_{i,n}| \leq Q_n^{1+o(1)} $, then the rational rank of the column vectors of the matrix with rows $ e_1, \ldots, e_k $ is at least $ k + \sum_{j=1}^k \tau_j $.
• Prove that the vectors $ e_1, \ldots, e_k $ are $ \mathbb{R} $-linearly independent and that their span avoids non-zero rational points.
• Derive a quantitative version of linear independence (conclusion (iii)) by showing that a certain convex body contains no non-zero integer point.
• Use a transference principle to relate the smallness of linear forms at points to the rank of the underlying vector system over $ \mathbb{Q} $.

Experimental results

Research questions

• RQ1 What is the minimal rational rank of a family of $ p $ vectors in $ \mathbb{R}^k $, given that there exists a sequence of integer-coefficient linear forms small at $ k $ distinct points?
• RQ2 How can Nesterenko's linear independence criterion for scalars be generalized to vector systems in higher-dimensional spaces?
• RQ3 What quantitative conditions on the decay rate of linear forms at $ k $ points ensure that the rational span of the corresponding vectors has high dimension?
• RQ4 Can the proof of irrationality results for zeta values at odd integers be simplified by reducing the number of required independent small forms?
• RQ5 Is it possible to derive lower bounds on the rank of a vector family without proving full linear independence, using only one sequence of small linear forms?

Key findings

• The paper establishes a generalization of Nesterenko's criterion to vector systems, proving that if $ k $ points in $ \mathbb{R}^p $ admit a sequence of integer-coefficient linear forms decaying as $ Q_n^{-\tau_j + o(1)} $, then the $ \mathbb{Q} $-rank of the column vectors of the matrix formed by these points is at least $ k + \sum_{j=1}^k \tau_j $.
• For any $ \varepsilon > 0 $ and sufficiently large odd $ a $, there exist $ N = \left\lfloor \frac{1 - \varepsilon}{1 + \log 2} \log a \right\rfloor $ odd integers $ \sigma_1, \ldots, \sigma_N \in [3, a] $ such that $ 1, \zeta(\sigma_1), \ldots, \zeta(\sigma_N) $ are linearly independent over $ \mathbb{Q} $, and the $ \sigma_i $ are separated by more than $ a^\varepsilon $.
• The method proves that the set $ \mathcal{C}(\varepsilon, Q) $, consisting of vectors within $ Q^{\tau_j - \varepsilon} $ in the span of $ e_1, \ldots, e_k $ and within $ Q^{-1 - \varepsilon} $ in the orthogonal complement, contains no non-zero integer point.
• The result implies that the $ \mathbb{R} $-span of $ e_1, \ldots, e_k $ contains no non-zero rational vector, ensuring strong linear independence over $ \mathbb{R} $.
• The criterion allows for alternative proofs of known linear independence results (e.g., for $ \zeta(2), \zeta(3), \log 2 $) using only one sequence of small forms instead of $ p $ independent ones.
• By applying the criterion to zeta values, the paper provides a framework to derive lower bounds on the $ \mathbb{Q} $-rank of families of zeta values without requiring full linear independence, even when the full set is not known to be independent.