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

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 C(G) for the maximum cardinality of such a set. We first show that C(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 C(mathbbZ/nmathbbZ)=varphi(n). We prove that liminfn→∞C(mathbbZ/nmathbbZ)/varphi(n)=1, whereas limsupn→∞C(mathbbZ/nmathbbZ)/varphi(n)=∞, 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]>dimK A for every nonzero a∈ A. Since this condition depends on dimK 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 C(L/K)=[L:K]-dmax(L/K), where dmax(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.