Arithmetic properties of Cantor sets involving non-diagonal forms
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915384951570432 |
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| author | Zhao, Haotian |
| author_facet | Zhao, Haotian |
| contents | We show conditions on $k$ such that any number $x$ in the interval $[0, k/2]$ can be represented in the form $x_1^{a_1} x_2^{a_2} + x_3^{a_3} x_4^{a_4} + \cdots + x_{k-1}^{a_{k-1}} x_k^{a_k}$, where the exponents $a_{2i-1}$ and $a_{2i}$ are positive integers satisfying $a_{2i-1} + a_{2i} = s$ for $i = 1, 2, \dots, k/2$, and each $x_i$ belongs to the generalized Cantor set. Moreover, we discuss different types of non-diagonal polynomials and clarify the optimal results in low-dimensional cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic properties of Cantor sets involving non-diagonal forms Zhao, Haotian Number Theory 28A80, 11P05 We show conditions on $k$ such that any number $x$ in the interval $[0, k/2]$ can be represented in the form $x_1^{a_1} x_2^{a_2} + x_3^{a_3} x_4^{a_4} + \cdots + x_{k-1}^{a_{k-1}} x_k^{a_k}$, where the exponents $a_{2i-1}$ and $a_{2i}$ are positive integers satisfying $a_{2i-1} + a_{2i} = s$ for $i = 1, 2, \dots, k/2$, and each $x_i$ belongs to the generalized Cantor set. Moreover, we discuss different types of non-diagonal polynomials and clarify the optimal results in low-dimensional cases. |
| title | Arithmetic properties of Cantor sets involving non-diagonal forms |
| topic | Number Theory 28A80, 11P05 |
| url | https://arxiv.org/abs/2503.03933 |