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

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.

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Number Theory (math.NT); Artificial Intelligence (cs.AI); Group Theory (math.GR)

arXiv:2607.24847 [math.NT]

(or arXiv:2607.24847v1 [math.NT] for this version)

https://doi.org/10.48550/arXiv.2607.24847 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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