The structure of sets with cube-avoiding sumsets
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915029596504064 |
|---|---|
| author | Karam, Thomas Keevash, Peter |
| author_facet | Karam, Thomas Keevash, Peter |
| contents | We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$ then $E_1, \dots, E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1' \times G^{I^c}, \dots, E_d' \times G^{I^c}$, where the size of $I \subset [n]$ is non-zero and independent of $n$, and $E_1',\dots,E_d'$ are subsets of $G^{I}$ such that the sumset $E_1'+\dots+E_d'$ avoids $Z_0^I$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The structure of sets with cube-avoiding sumsets Karam, Thomas Keevash, Peter Combinatorics Number Theory We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$ then $E_1, \dots, E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1' \times G^{I^c}, \dots, E_d' \times G^{I^c}$, where the size of $I \subset [n]$ is non-zero and independent of $n$, and $E_1',\dots,E_d'$ are subsets of $G^{I}$ such that the sumset $E_1'+\dots+E_d'$ avoids $Z_0^I$. |
| title | The structure of sets with cube-avoiding sumsets |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2411.14145 |