Measure doubling in unimodular locally compact groups and quotients
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| Format: | Preprint |
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2024
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| _version_ | 1866929605792759808 |
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| author | Kong, Zuxiang Peng, Fei Tran, Chieu-Minh |
| author_facet | Kong, Zuxiang Peng, Fei Tran, Chieu-Minh |
| contents | We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $μ_G$, and a compact $A\subseteq G$ of positive measure with $μ_G(A^2)\leq Kμ_G(A)$. Let $H$ be a closed normal subgroup of G and $π: G \rightarrow G/H$ be the quotient map. With the further assumption that $A= A^{-1}$, we show $$μ_{G/H}(πA ^2) \leq K^2 μ_{G/H}(πA).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-ε)K^2$ for any $ε>0$.
In the general case (without $A=A^{-1}$), we show $μ_{G/H}(πA ^2) \leq K^3 μ_{G/H}(πA)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with $μ_G(B)> μ_G(A)/2$ such that $ μ_{G/H}(πB^2) < 2K μ_{G/H}(πB)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure doubling in unimodular locally compact groups and quotients Kong, Zuxiang Peng, Fei Tran, Chieu-Minh Group Theory Combinatorics Logic 22D05 (Primary) 51F99, 22E30, 03C99, 11B30 (Secondary) We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $μ_G$, and a compact $A\subseteq G$ of positive measure with $μ_G(A^2)\leq Kμ_G(A)$. Let $H$ be a closed normal subgroup of G and $π: G \rightarrow G/H$ be the quotient map. With the further assumption that $A= A^{-1}$, we show $$μ_{G/H}(πA ^2) \leq K^2 μ_{G/H}(πA).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-ε)K^2$ for any $ε>0$. In the general case (without $A=A^{-1}$), we show $μ_{G/H}(πA ^2) \leq K^3 μ_{G/H}(πA)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with $μ_G(B)> μ_G(A)/2$ such that $ μ_{G/H}(πB^2) < 2K μ_{G/H}(πB)$. |
| title | Measure doubling in unimodular locally compact groups and quotients |
| topic | Group Theory Combinatorics Logic 22D05 (Primary) 51F99, 22E30, 03C99, 11B30 (Secondary) |
| url | https://arxiv.org/abs/2411.17246 |