Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist

2026-07-24

AI Summary

This study investigates the maximal size of Chowla sets in finite groups and their natural linear analogue in finite field extensions. By establishing a precise connection between Chowla sets and the distribution of element orders in groups, the work introduces for the first time a linear-algebraic generalization—Chowla subspaces. Leveraging tools from group theory, field theory, invariant factor decomposition, and normal basis constructions, and aided by the human–AI collaborative reasoning platform Co-Scientist, the authors derive explicit formulas for Chowla sets in cyclic groups and finite abelian p-groups, characterize their asymptotic upper and lower limits, and obtain closed-form expressions for the dimensions of Chowla subspaces in finite separable field extensions.

Abstract

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $\Ccal(G)$ for the maximum cardinality of such a set. We first show that $\Ccal(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $\Ccal(\mathbb Z/n\mathbb Z)=\varphi(n)$. We prove that $\liminf_{n\to\infty}\Ccal(\mathbb Z/n\mathbb Z)/\varphi(n)=1$, whereas $\limsup_{n\to\infty}\Ccal(\mathbb Z/n\mathbb Z)/\varphi(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $\Ccal(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human--AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.

Problem

• Chowla set
• extremal invariant
• finite group
• field extension
• linear analogue

Innovation

• Chowla sets
• extremal invariant
• finite groups
• Chowla subspaces
• human-AI collaboration