AI 中文总结
该研究在数论猜想下，证明单位圆幂函数图像扩张实域时，开可定义集仍为半代数集，通过双排序结构与Hrushovski合并法构造了新的实域驯服扩张实例。
AI 中文摘要
我们研究单位圆上幂函数图像对实域的扩张。在一个自然的数论猜想下，我们证明添加这类稠密子集不会增加可定义集的拓扑复杂度：每个开可定义集仍保持半代数性。该证明采用双排序结构，分离线性与代数数据，灵感源自Zilber的幂运算。运用Hrushovski的合并方法，我们构造并公理化一类丰富结构，随后证明目标结构是其模型，这为实域通过稠密轨迹的驯服扩张提供了新实例。
英文摘要
We study the expansion of the real field by the graphs of power functions on the unit circle. Under a natural number-theoretic conjecture, we prove that adding such dense subsets does not increase the topological complexity of definable sets: every open definable set remains semialgebraic. The proof uses a two-sorted structure that separates the linear and algebraic data, inspired by Zilber's raising to powers. Using Hrushovski's amalgamation method, we construct and axiomatize a class of rich structures, and then show that the intended structure is a model. This provides a new example of a tame expansion of the real field by dense trajectories.
