Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$
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| Format: | Preprint |
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2023
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| _version_ | 1866913643191336960 |
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| author | Péringuey, Paul de Roton, Anne |
| author_facet | Péringuey, Paul de Roton, Anne |
| contents | We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of integers and when $A$ is a bounded subset of real numbers. More precisely we give a sharp upper bound for the density of $A$ in $[\inf(A),x]$ for $x\le\sup(A)$, and construct sets reaching this density for any given $x$ in this range. This gives some information on sets involved in the structural description of some critical sets in Freiman's $3k-4$ theorem in both cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_13891 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$ Péringuey, Paul de Roton, Anne Number Theory Combinatorics 11P70 We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of integers and when $A$ is a bounded subset of real numbers. More precisely we give a sharp upper bound for the density of $A$ in $[\inf(A),x]$ for $x\le\sup(A)$, and construct sets reaching this density for any given $x$ in this range. This gives some information on sets involved in the structural description of some critical sets in Freiman's $3k-4$ theorem in both cases. |
| title | Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$ |
| topic | Number Theory Combinatorics 11P70 |
| url | https://arxiv.org/abs/2311.13891 |