Exceptional set estimate through Brascamp-Lieb inequality

Abstract

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 $k\le \frac{n}{2}$, $\beta \in (0,1]$ and $\gamma \in (\beta ,\frac{k}{n}(1+\beta )]$, we have \[T(1+β,γ)=k(n-k)-k.\] When $k\ge \frac{n}{2}$, $\beta \in (0,1]$ and $\gamma \in (\beta ,(1-\frac{k}{n})+\frac{k}{n}\beta ]$, we have \[T(n-1+β,k-1+γ)=k(n-k)-(n-k).\]

28 pages, journal version