2021/01/10 by Tyulenev, Alexander
#28A12 #28A78 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2101.03556
Let S ⊂ ℝn be a nonempty set. Given d ∈ [0,n) and a cube Q ⊂ ℝn with l=l(Q) ∈ (0,1], we show that if the d-Hausdorff content Hd∞(Q ∩ S) < λld for some λ ∈ (0,1), then the set Q ∖ S contains a specific cavity. More precisely, we prove existence of a pseudometric ρ=ρS,d such that for each sufficiently small δ> 0 the δ-neighborhood Uρδl(S) of S in the pseudometric ρ does not contain the whole Q. Moreover, we establish the existence of constants δ=δ(n,d,λ)>0 and \underlineγ=\underlineγ(n,d,λ)>0 such that Ln(Q ∖ Uρδl(S)) ≥ \underlineγ ln for all δ∈ (0,δ). If, in addition, the set S is d-lower content regular, we prove existence of a constant \underlineτ=\underlineτ(n,d,λ)>0 such that the cube Q is \underlineτ-porous. The sharpness of the results is illustrated by several examples.