The Density Finite Sums Theorem
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909787654979584 |
|---|---|
| author | Kra, Bryna Moreira, Joel Richter, Florian K. Robertson, Donald |
| author_facet | Kra, Bryna Moreira, Joel Richter, Florian K. Robertson, Donald |
| contents | For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$ for all $m\in\{1,\ldots,k\}$. This can be viewed as a density analog of Hindman's finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Density Finite Sums Theorem Kra, Bryna Moreira, Joel Richter, Florian K. Robertson, Donald Dynamical Systems Combinatorics Number Theory 37A05 37A44 37B20 05D10 For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$ for all $m\in\{1,\ldots,k\}$. This can be viewed as a density analog of Hindman's finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups. |
| title | The Density Finite Sums Theorem |
| topic | Dynamical Systems Combinatorics Number Theory 37A05 37A44 37B20 05D10 |
| url | https://arxiv.org/abs/2504.06424 |