Sumset size races for measurable sets
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915735815585792 |
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| author | Nathanson, Melvyn B. |
| author_facet | Nathanson, Melvyn B. |
| contents | Let $G$ be a locally compact abelian group with Haar measure $μ$. For integers $n \geq 2$ and $H \geq 2$ and for any $n$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_H \in \mathbf{N}^n$, there exist measurable subsets $A_1,\ldots, A_n$ of $G$ such that the $n$-tuple $\left( μ(hA_1),\ldots, μ(hA_n) \right)$ has the same relative order as the $n$-tuple $\mathbf{u}_h$ for all $h = 1,\ldots, H$.
For integers $m_{i,h}$ for $i =1,\ldots, n-1$ and $h = 1,\ldots, H$, there are Lebesgue measurable sets $A_1,\ldots, A_n$ in $\mathbf{R}$ such that $μ(hA_{i+1}) - μ(hA_i) = m_{i,h}$ for all $i$ and $h$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sumset size races for measurable sets Nathanson, Melvyn B. Number Theory 11B13, 11B05, 11B75, 11P70, 22D99 Let $G$ be a locally compact abelian group with Haar measure $μ$. For integers $n \geq 2$ and $H \geq 2$ and for any $n$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_H \in \mathbf{N}^n$, there exist measurable subsets $A_1,\ldots, A_n$ of $G$ such that the $n$-tuple $\left( μ(hA_1),\ldots, μ(hA_n) \right)$ has the same relative order as the $n$-tuple $\mathbf{u}_h$ for all $h = 1,\ldots, H$. For integers $m_{i,h}$ for $i =1,\ldots, n-1$ and $h = 1,\ldots, H$, there are Lebesgue measurable sets $A_1,\ldots, A_n$ in $\mathbf{R}$ such that $μ(hA_{i+1}) - μ(hA_i) = m_{i,h}$ for all $i$ and $h$. |
| title | Sumset size races for measurable sets |
| topic | Number Theory 11B13, 11B05, 11B75, 11P70, 22D99 |
| url | https://arxiv.org/abs/2601.11490 |