Note on a conjecture of Sárközy on special sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908696270864384 |
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| author | Ding, Yuchen Li, Huixi Zhang, Zihan |
| author_facet | Ding, Yuchen Li, Huixi Zhang, Zihan |
| contents | Let $α>1$ be an irrational number and $k\ge 2$ a positive integer. Let $f(x)$ be a polynomial with positive integer coefficients. Solving a 2001 problem of Sárközy on special sequences, Hegyvári proved in 2003 that there exists an infinite sequence $A$ with density $\frac{1}{k}-\frac{1}{kα}$ such that $$ \big\{f(a_1)+\ldots+f(a_k): a_i\in A, 1\le i\le k\big\}\cap \big\{\lfloor nα\rfloor: n\in \mathbb{N}\big\}=\emptyset. $$ Hegyvári also proved that the density given by him is optimal for $k=2$. In this article, we show that the density $\frac{1}{k}-\frac{1}{kα}$ given by Hegyvári is actually optimal for all $k\ge 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_25025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on a conjecture of Sárközy on special sequences Ding, Yuchen Li, Huixi Zhang, Zihan Number Theory Let $α>1$ be an irrational number and $k\ge 2$ a positive integer. Let $f(x)$ be a polynomial with positive integer coefficients. Solving a 2001 problem of Sárközy on special sequences, Hegyvári proved in 2003 that there exists an infinite sequence $A$ with density $\frac{1}{k}-\frac{1}{kα}$ such that $$ \big\{f(a_1)+\ldots+f(a_k): a_i\in A, 1\le i\le k\big\}\cap \big\{\lfloor nα\rfloor: n\in \mathbb{N}\big\}=\emptyset. $$ Hegyvári also proved that the density given by him is optimal for $k=2$. In this article, we show that the density $\frac{1}{k}-\frac{1}{kα}$ given by Hegyvári is actually optimal for all $k\ge 2$. |
| title | Note on a conjecture of Sárközy on special sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.25025 |