EXCEPTIONAL POINTS FOR DENSITIES GENERATED BY SEQUENCES

Abstract

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every measurable set $A\subset \mathrm{ℝ}$ with and , there is a point at which both the lower densities of $A$ and of the complement of A are at least $\delta$. The problem of determining the supremum ${\delta }_{\mathcal{H}}$ of possible values of this $\delta$ 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 $\lambda (A\cap [x-h,x+h\left]\right)/2h$. Replacing $\lambda (A\cap [x-h,x+h\left]\right)/2h$ by $\lambda (A\cap [x-t_n,x+t_n\left]\right)/2t_n$ for a fixed decreasing sequence $⟨t⟩$ tending to zero, we obtain a definition of the constant ${\delta }_{⟨t⟩}$. In our paper we look for an upper bound of all such constants.