Exceptional set estimate through Brascamp-Lieb inequality
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917585910497280 |
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| author | Gan, Shengwen |
| author_facet | Gan, Shengwen |
| contents | Fix integers $1\le k<n$, and numbers $a,s$ satisfying $0<s<\min\{k,a\}$. The problem of exceptional set estimate is to determine \[T(a,s):=\sup_{A\subset \mathbb{R}^n\ \text{dim}A=a}\text{dim}(\{ V\in G(k,n): \text{dim}(π_V(A))<s \}). \] In this paper, we prove a new upper bound for $T(a,s)$ by using Brascamp-Lieb inequality. As one of the corollary, we obtain the estimate \[T(a,\frac{k}{n}a)\le k(n-k)-\min\{k,n-k\}, \] which improves a previous result $T(a,\frac{k}{n}a)\le k(n-k)-1$ of He.
By constructing examples, we can determine the explicit value of $T(a,s)$ for certain $(a,s)$:
When $k\le \frac{n}{2}$, $β\in(0,1]$ and $γ\in(β,\frac{k}{n}(1+β)]$, we have \[T(1+β,γ)=k(n-k)-k.\]
When $k\ge \frac{n}{2}$, $β\in(0,1]$ and $γ\in (β, (1-\frac{k}{n})+\frac{k}{n}β]$, we have \[T(n-1+β,k-1+γ)=k(n-k)-(n-k).\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07675 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exceptional set estimate through Brascamp-Lieb inequality Gan, Shengwen Classical Analysis and ODEs Combinatorics Fix integers $1\le k<n$, and numbers $a,s$ satisfying $0<s<\min\{k,a\}$. The problem of exceptional set estimate is to determine \[T(a,s):=\sup_{A\subset \mathbb{R}^n\ \text{dim}A=a}\text{dim}(\{ V\in G(k,n): \text{dim}(π_V(A))<s \}). \] In this paper, we prove a new upper bound for $T(a,s)$ by using Brascamp-Lieb inequality. As one of the corollary, we obtain the estimate \[T(a,\frac{k}{n}a)\le k(n-k)-\min\{k,n-k\}, \] which improves a previous result $T(a,\frac{k}{n}a)\le k(n-k)-1$ of He. By constructing examples, we can determine the explicit value of $T(a,s)$ for certain $(a,s)$: When $k\le \frac{n}{2}$, $β\in(0,1]$ and $γ\in(β,\frac{k}{n}(1+β)]$, we have \[T(1+β,γ)=k(n-k)-k.\] When $k\ge \frac{n}{2}$, $β\in(0,1]$ and $γ\in (β, (1-\frac{k}{n})+\frac{k}{n}β]$, we have \[T(n-1+β,k-1+γ)=k(n-k)-(n-k).\] |
| title | Exceptional set estimate through Brascamp-Lieb inequality |
| topic | Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2308.07675 |