An analytic version of stable arithmetic regularity
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929387674271744 |
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| author | Conant, Gabriel Pillay, Anand |
| author_facet | Conant, Gabriel Pillay, Anand |
| contents | We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary function $f(x\cdot y)$ is stable in the sense of continuous logic. Roughly speaking, our main result says that if $G$ is amenable, then any stable function on $G$ is almost constant on all translates of a unitary Bohr neighborhood in $G$ of bounded complexity. The proof uses ingredients from topological dynamics and continuous model theory. We also prove several applications which generalize results in arithmetic combinatorics to nonabelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An analytic version of stable arithmetic regularity Conant, Gabriel Pillay, Anand Logic Combinatorics Group Theory We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary function $f(x\cdot y)$ is stable in the sense of continuous logic. Roughly speaking, our main result says that if $G$ is amenable, then any stable function on $G$ is almost constant on all translates of a unitary Bohr neighborhood in $G$ of bounded complexity. The proof uses ingredients from topological dynamics and continuous model theory. We also prove several applications which generalize results in arithmetic combinatorics to nonabelian groups. |
| title | An analytic version of stable arithmetic regularity |
| topic | Logic Combinatorics Group Theory |
| url | https://arxiv.org/abs/2401.14363 |