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arXiv · 2608.08655

Raising to powers on the unit circle

Yilong Zhang
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Abstract

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.

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Yilong Zhang. 2026-08-09. Raising to powers on the unit circle. https://arxiv.org/abs/2608.08655

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