Uniform sets with few progressions via colorings
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913901749207040 |
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| author | Deng, Mingyang Tidor, Jonathan Zhao, Yufei |
| author_facet | Deng, Mingyang Tidor, Jonathan Zhao, Yufei |
| contents | Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $α$ and 4-term arithmetic progression (4-AP) density at most $α^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $α$ and 4-AP density at most $α^{4+c}$ for some small constant $c>0$. We show that an affirmative answer to Ruzsa's question would follow from the existence of an $N^{o(1)}$-coloring of $[N]$ without symmetrically colored 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa's question is equivalent to our arithmetic Ramsey question.
We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $α$ and $k$-AP density at most $α^{c_k \log(1/α)}$. We also prove generalizations to arbitrary one-dimensional patterns. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06914 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform sets with few progressions via colorings Deng, Mingyang Tidor, Jonathan Zhao, Yufei Combinatorics Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $α$ and 4-term arithmetic progression (4-AP) density at most $α^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $α$ and 4-AP density at most $α^{4+c}$ for some small constant $c>0$. We show that an affirmative answer to Ruzsa's question would follow from the existence of an $N^{o(1)}$-coloring of $[N]$ without symmetrically colored 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa's question is equivalent to our arithmetic Ramsey question. We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $α$ and $k$-AP density at most $α^{c_k \log(1/α)}$. We also prove generalizations to arbitrary one-dimensional patterns. |
| title | Uniform sets with few progressions via colorings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.06914 |