Spread Furstenberg Sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929756793995264 |
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| author | Bright, Paige Dhar, Manik |
| author_facet | Bright, Paige Dhar, Manik |
| contents | We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal G(n,k)$ such that for all $P\in \mathcal P$, there exists a translation vector $a_P \in \mathbb{R}^n$ such that $\dim(F\cap (P + a_P)) \geq s$. We show that given $k \geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;k)$-spread Furstenberg set $F$ in $\mathbb{R}^n$ satisfies \[ \dim F \geq n-k + s - \frac{k(n-k) - t}{\lceil s\rceil - k_0 +1 }. \] Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spread Furstenberg Sets Bright, Paige Dhar, Manik Classical Analysis and ODEs Metric Geometry 28A75, 28A78 We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal G(n,k)$ such that for all $P\in \mathcal P$, there exists a translation vector $a_P \in \mathbb{R}^n$ such that $\dim(F\cap (P + a_P)) \geq s$. We show that given $k \geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;k)$-spread Furstenberg set $F$ in $\mathbb{R}^n$ satisfies \[ \dim F \geq n-k + s - \frac{k(n-k) - t}{\lceil s\rceil - k_0 +1 }. \] Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields. |
| title | Spread Furstenberg Sets |
| topic | Classical Analysis and ODEs Metric Geometry 28A75, 28A78 |
| url | https://arxiv.org/abs/2412.18193 |