A sharp upper bound for the Hausdorff dimension of the set of exceptional points for the strong density theorem

Given an E⊆RmE⊆Rm, Lebesgue measurable, we construct a real function ψ:R+→R+ψ:R+→R+ (depending on E  ) increasing, with limt→0+⁡ψ(t)=0 such thatlimx∈Rd(R)→0⁡|R∩Ec||R|⋅ψ(d(R))=0fora.e.x∈E (where R   is an interval in RmRm and d stands for the diameter). This gives a new constructive proof of a problem posed by S.J. Taylor (1959) [7, p. 314]. Furthermore, the constructive method we use, gives a sharp upper bound for the Hausdorff dimension of the set of exceptional points, for the strong density theorem of Saks.