[Paper Review] A Note on Combinatorial Derivation

Joshua Erde|arXiv (Cornell University)|Oct 29, 2012

Limits and Structures in Graph Theory2 references4 citations

TL;DR

This paper resolves several open questions posed by Protasov concerning combinatorial derivation in infinite groups. It proves that in any finite partition of a large set, at least one part has a combinatorial derivation that is itself large, and establishes that every ∇-thin set is sparse. The results are derived using structural group-theoretic arguments and cardinality-based constructions, with key bounds on the size of finite sets needed to cover the group via the derivation operator.

ABSTRACT

Given an infinite group $G$ and a subset $A$ of $G$ we let $Δ(A) = {g \in G : |gA \cap A| =\infty}$ (this is sometimes called the combinatorial derivation of $A$). A subset $A$ of $G$ is called large if there exists a finite subset $F$ of $G$ such that $FA=G$. We show that given a large set $X$, and a decomposition $X=A_1 \cup ... \cup A_n$, there must exist an $i$ such that $Δ(A_i)$ is large. This answers a question of Protasov. We also answer a number of related questions of Protasov.

Motivation & Objective

• To resolve open questions posed by Protasov regarding the combinatorial derivation Δ(A) in infinite groups.
• To determine whether large sets are necessarily Δ-large, and whether finite partitions of large sets must contain a Δ-large part.
• To investigate the relationship between ∇-thin sets and sparsity, and to clarify the structure of sets with infinite intersection properties under translation.
• To establish bounds on the size of finite sets F such that FΔ(A_i) = G in finite partitions, addressing uniformity across group types.

Proposed method

• Uses the concept of cofinite translates: if FA is cofinite, then FΔ(A) = G, which implies Δ-largeness.
• Applies maximal finite sets F such that |f_iA ∩ f_jA| < ∞ to show non-almost P-small sets are Δ-large.
• Constructs infinite subsets Y ⊆ A such that Δ(Y) = X for any countable X ⊆ Δ(A), using infinite chains of intersecting translates.
• Employs cardinality arguments in large groups (e.g., (Z_2)^κ) to show that uncountable subsets of Δ(A) may not arise as Δ(Y) for any Y ⊆ A.
• Uses the identity Δ(FA) = FΔ(A)F⁻¹ to relate translates and derivation sets.
• Proves that non-sparse sets have infinite Δ^n(A) for all n, showing they cannot be ∇-thin.

Experimental results

Research questions

• RQ1 Is every large subset of an infinite group Δ-large?
• RQ2 Does there exist a function f: N → N such that in any n-partition of a group G, one part A_i satisfies G = FΔ(A_i) for some F with |F| ≤ f(n)?
• RQ3 Given a symmetric partition G = A₁ ∪ … ∪ A_n with e ∈ A_i, does some A_i contain an infinite X with Δ(X) ⊆ A_i?
• RQ4 Is every ∇-thin set necessarily sparse?
• RQ5 Can the derivation operator Δ generate uncountable sets as Δ(Y) for Y ⊆ A?

Key findings

• Every large set in an infinite group has a part A_i in any finite partition such that Δ(A_i) is large, answering Question A positively.
• For any finite partition of a set X with FX cofinite, there exists an i and a finite F' with |F'| ≤ |F|(|F|+1)^{2^{n-1}-1} such that F'Δ(A_i) = G, providing a uniform bound for Question B.
• For any countable symmetric X ⊆ Δ(A) with e ∈ X, there exists Y ⊆ A such that Δ(Y) = X, resolving Question C positively.
• Every ∇-thin set is sparse, providing a positive answer to Question D.
• There exist uncountable subsets X ⊆ Δ(A) for which no Y ⊆ A satisfies Δ(Y) = X, showing the necessity of countability in Theorem 2.
• The group (Z, +) admits a 2-sparse set that is not ∇-thin, demonstrating that the converse of the sparse ⇒ ∇-thin implication does not hold.