AI 中文总结
针对有限域乘法子群，研究其与广义算术级数、和集分解的关系，得出真子群为广义算术级数的充要条件，及加法不可约、和差表示数的相关结论。
AI 中文摘要
设q = p^f，A是乘法子群且满足F_p(A) = F_q。我们证明：真子群A为广义算术级数（GAP）当且仅当|A| ∈ {1,2,4}，并确定全群F_q^×何时为GAP。对某些子群族，我们得到更强结论：A是加法不可约的。特别地，若|A|>4且存在e≥1使得p^e ≡ -1 mod |A|，则A不存在非平凡和集分解。我们还证明：当[F_q^×:A]≥3且|A|≥5时，每个非零c作为A中两个元素的和（或差）的表示数少于|A|/2，这一结果或有独立意义。
英文摘要
Let $q=p^f$, and let $A\leq\mathbb{F}_q^\times$ be a multiplicative subgroup with $\mathbb{F}_p(A)=\mathbb{F}_q$. We prove that a proper subgroup $A$ is a generalized arithmetic progression (GAP) if and only if $|A| \in \{1, 2, 4\}$, and we determine when the full group $\mathbb{F}_q^\times$ is a GAP. For certain families of subgroups, we obtain the stronger conclusion that $A$ is additively irreducible. In particular, if $|A|>4$ and $p^e\equiv-1\pmod{|A|}$ for some $e\ge1$, then $A$ admits no nontrivial sumset decomposition. We also prove that every $c \neq 0$ has fewer than $|A|/2$ representations as a sum (or difference) of two elements of $A$ whenever $[\mathbb{F}_q^\times:A] \ge3$ and $|A| \ge 5$, which may be of independent interest.
Comments16 pages
