Exceptional set estimate through Brascamp-Lieb inequality

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 $$T(a,s):=\underset{A\subset {\mathbb{R}}^n\text{ dim}A=a}{\sup }\text{dim}(\{V\in G(k,n):\text{dim}({\pi }_V(A))<s\})\mathrm{.}$$ 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 $$T(1+\beta ,\gamma )=k(n-k)-k\mathrm{.}$$ When $k\ge \frac{n}{2}$, $\beta \in (0,1]$ and $\gamma \in (\beta ,(1-\frac{k}{n})+\frac{k}{n}\beta ]$, we have $$T(n-1+\beta ,k-1+\gamma )=k(n-k)-(n-k)\mathrm{.}$$