There are infinitely many Hilbert cubes of dimension 3 in the set of squares
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908941887209472 |
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| author | Bremner, Andrew Elsholtz, Christian Ulas, Maciej |
| author_facet | Bremner, Andrew Elsholtz, Christian Ulas, Maciej |
| contents | A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erdős and Freedman asked whether the maximal dimension of a Hilbert cube in the set $\cal{S}=\{n^2:\;n\in\mathbb{N}\}$ of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if $H(a_{0}; a_{1}, \ldots, a_{d})\subset \cal{S}\cap [0, N]$, then $d\leq 7 \log\log N$ for all sufficiently large values of $N$. Here we prove that there exist at least $\gg N^{1/8}$ Hilbert cubes $H(a_{0}; a_{1}, a_{2}, a_{3})$ with $a_{0}, a_{1}, a_{2}, a_{3}\in [0,N]$ in the set of squares. Moreover, we prove that for each $i, j\in\{0, 1, 2, 3\}$ with $i<j$, the set $$ \left\{\frac{a_{i}}{a_{j}}:\;H(a_{0}; a_{1}, a_{2}, a_{3})\subset S\right\} $$ is dense in the set of positive real numbers (in the Euclidean topology). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05459 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | There are infinitely many Hilbert cubes of dimension 3 in the set of squares Bremner, Andrew Elsholtz, Christian Ulas, Maciej Number Theory Combinatorics A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erdős and Freedman asked whether the maximal dimension of a Hilbert cube in the set $\cal{S}=\{n^2:\;n\in\mathbb{N}\}$ of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if $H(a_{0}; a_{1}, \ldots, a_{d})\subset \cal{S}\cap [0, N]$, then $d\leq 7 \log\log N$ for all sufficiently large values of $N$. Here we prove that there exist at least $\gg N^{1/8}$ Hilbert cubes $H(a_{0}; a_{1}, a_{2}, a_{3})$ with $a_{0}, a_{1}, a_{2}, a_{3}\in [0,N]$ in the set of squares. Moreover, we prove that for each $i, j\in\{0, 1, 2, 3\}$ with $i<j$, the set $$ \left\{\frac{a_{i}}{a_{j}}:\;H(a_{0}; a_{1}, a_{2}, a_{3})\subset S\right\} $$ is dense in the set of positive real numbers (in the Euclidean topology). |
| title | There are infinitely many Hilbert cubes of dimension 3 in the set of squares |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2604.05459 |