Abstract
We prove structure theorems for o-minimal definable subsets \(S\subset G\) of definable groups containing large multiplicative structures, and show definable groups do not have bounded torsion arbitrarily close to the identity. As an application, for certain models of n-step random walks X in G we show upper bounds \(\mathbb {P}(X\in S)\le n^{-C}\) and a structure theorem for the steps of X when \(\mathbb {P}(X\in S)\ge n^{-C'}\).
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Notes
Technically, the scope of Theorem A is only for o-minimal expansions of \({\mathbb {R}}_{an,\exp }\), which does not contain all o-minimal expansions of \({\mathbb {R}}_{alg}\) as covered by Theorem 2.2. Theorem E, which does include o-minimal expansions of \({\mathbb {R}}_{alg}\), has the condition of avoiding a line segment in \((\mathbb {R}^d,+)\) replaced with the stronger condition of not containing unboundedly large arithmetic progressions, but fortunately, these latter conditions are in fact equivalent in \((\mathbb {R}^d,+)\) by the Uniform Finiteness Principle Fact 5.1 applied to the definable family of subsets \(\{S\cap \{y+tv:t\in \mathbb {R}\}\}_{y\in \mathbb {R}^d,v\in \mathbb {R}^d{\setminus } 0}\) of \(\mathbb {R}^d\).
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Acknowledgements
I would like to thank Artem Chernikov for helpful comments, Matthew Kwan for helpful remarks on the exposition, Isaac Goldbring for pointing out an example of a definable group not satisfying (NSS), and Thomas Scanlon for additional references.
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