Asymmetric infinite sumsets in large sets of integers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918297022234624 |
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| author | Kousek, Ioannis |
| author_facet | Kousek, Ioannis |
| contents | We show that for any set $A \subset \mathbb{N}$ with positive upper density and any $\ell,m \in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1 \leq b_2 \}$, for infinite $B\subset \mathbb{N}$ and prove that any set $A\subset \mathbb{N}$ with lower density $\underline{d}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03112 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymmetric infinite sumsets in large sets of integers Kousek, Ioannis Dynamical Systems Combinatorics Number Theory 37A05, 05D10, 11B13, 11B30 We show that for any set $A \subset \mathbb{N}$ with positive upper density and any $\ell,m \in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1 \leq b_2 \}$, for infinite $B\subset \mathbb{N}$ and prove that any set $A\subset \mathbb{N}$ with lower density $\underline{d}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed. |
| title | Asymmetric infinite sumsets in large sets of integers |
| topic | Dynamical Systems Combinatorics Number Theory 37A05, 05D10, 11B13, 11B30 |
| url | https://arxiv.org/abs/2502.03112 |