EXCEPTIONAL POINTS FOR DENSITIES GENERATED BY SEQUENCES

Abstract

In spite of the Lebesgue density theorem, there is a positive δ such that, for every measurable set A⊂ℝ with λ(A)>0 and λ(ℝ\A)>0, there is a point at which both the lower densities of A and of the complement of A are at least δ. The problem of determining the supremum δH of possible values of this δ was studied by V. I. Kolyada, A. Szenes and others, and it was solved by O. Kurka. Lower density of A at x is defined as a lower limit of λ(A∩[x-h,x+h])/2h. Replacing λ(A∩[x-h,x+h])/2h by λ(A∩[x-tn,x+tn])/2tn for a fixed decreasing sequence ⟨t⟩ tending to zero, we obtain a definition of the constant δ⟨t⟩. In our paper we look for an upper bound of all such constants.