[Paper Review] On the distribution of multiplicatively dependent vectors

Konyagin, Sergei V., Min Sha+2 more|arXiv (Cornell University)|Mar 23, 2019

Analytic Number Theory Research22 references4 citations

TL;DR

This paper investigates the distribution of multiplicatively dependent vectors in real and complex spaces, proving that such vectors are dense in ℝⁿ and ℂⁿ when coordinates come from dense, powering-closed subsets like ℚ or number fields. Despite having zero Lebesgue measure, these sets are dense due to algebraic structure, and the paper introduces a refined covering radius to analyze irregularities in discrete cases like ℤⁿ and rings of integers in imaginary quadratic fields.

ABSTRACT

In this paper, we study the distribution of multiplicatively dependent vectors. For example, although they have zero Lebesgue measure, they are everywhere dense both in $\mathbb{R}^n$ and $\mathbb{C}^n$. We also study this property in a more detailed manner by considering the covering radius of such vectors.

Motivation & Objective

• To understand the distribution of multiplicatively dependent vectors in ℝⁿ and ℂⁿ, particularly when coordinates are drawn from number fields or rings of integers.
• To resolve the apparent paradox that these vectors have zero Lebesgue measure yet are dense in Euclidean space.
• To analyze irregularities in distribution for discrete sets like ℤⁿ and 𝒪_K when K is an imaginary quadratic field.
• To introduce a refined covering radius concept to quantify gaps in the distribution of such vectors in discrete settings.

Proposed method

• Prove density of 𝒮ₙ(S) in ℝⁿ and ℂⁿ for dense, powering-closed subsets S of ℝ or ℂ, using topological and algebraic arguments.
• Use the structure of multiplicative dependence via integer exponent vectors (k₁,…,kₙ) satisfying v₁^{k₁}⋯vₙ^{kₙ} = 1.
• Apply a constructive path-following argument on prime factorizations to find non-trivial solutions with small exponents in {-1,0,1}.
• Construct explicit counterexamples using tuples like (2, 2³, ..., 2^{3^{n-1}}) to show that not all sets yield dense images.
• Introduce a refined covering radius to measure maximal distance from any point in space to the nearest multiplicatively dependent vector.
• Analyze the distribution in ℤⁿ and rings of integers in imaginary quadratic fields where standard density fails.

Experimental results

Research questions

• RQ1 Under what conditions is the set of multiplicatively dependent vectors dense in ℝⁿ or ℂⁿ?
• RQ2 Why do multiplicatively dependent vectors—though measure zero—remain dense in ℝⁿ and ℂⁿ?
• RQ3 How can the distribution of such vectors be irregular in discrete rings like ℤ and 𝒪_K for imaginary quadratic fields?
• RQ4 Can a refined covering radius concept detect and quantify these irregularities in discrete settings?
• RQ5 What role does the powering-closure of the underlying set play in determining density of the resulting vector set?

Key findings

• The set of multiplicatively dependent vectors with coordinates in ℚ is dense in ℝⁿ for all n ≥ 2.
• For any number field K with [K:ℚ] ≥ 2 and 𝒪_K ∩ ℝ ≠ ℤ, the set 𝒮ₙ(𝒪_K ∩ ℝ) is dense in ℝⁿ.
• If K ⊊ ℝ and [K:ℚ] ≥ 3, then 𝒮ₙ(K) is dense in ℂⁿ.
• The set 𝒮ₙ(ℤ) is not dense in ℝⁿ, and the set 𝒮ₙ(𝒪_K) for imaginary quadratic K is not dense in ℂⁿ.
• A refined covering radius concept reveals significant irregularities in the distribution of 𝒮ₙ(ℤ) and 𝒮ₙ(𝒪_K) in their respective spaces.
• There exist explicit tuples (α₁,…,αₙ) in ℝⁿ or ℂⁿ such that no neighborhood contains any multiplicatively dependent vector, proving non-density for certain sets.