Exceptional Set Estimate Through Brascamp–Lieb Inequality

Fix integers 1≤ k<n, and numbers $a,s$ satisfying 0<s<min ,a\. The problem of exceptional set estimate is to determine align* & T(a,s):=_A⊂ Rⁿ\ dimA=adim(\ V∈ G(k,n): dim(πV(A))<s \). align*In this paper, we prove a new upper bound for $T(a,s)$ by using Brascamp–Lieb inequality. As one of the corollaries, we obtain the estimate align* & T(a,k/na)≤ k(n-k)-min,n-k\, align*which improves a previous result T(a,k/na)≤ 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≤ n/2, β ∈ (0,1] and γ ∈ (β ,k/n(1+β )], we have align* &T(1+β,γ)=k(n-k)-k.align* When k≥ n/2, β ∈ (0,1] and γ ∈ (β , (1-k/n)+k/nβ ], we have align* &T(n-1+β,k-1+γ)=k(n-k)-(n-k).align*