Normal numbers and subsets of N with given densities

@inproceedings{MATHEMATICAE2007NormalNA,
title={Normal numbers and subsets of N with given densities},
author={FUNDAMENTA MATHEMATICAE},
year={2007}
}

FUNDAMENTA MATHEMATICAE

Published 2007

For X ⊆ [0, 1], let DX denote the collection of subsets of N whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of DX . For α ≥ 3, X is properly Dξ(Π α) iff DX is properly Dξ(Π 1+α). We also show that for every nonempty set X ⊆ [0, 1], DX is Π0 3 -hard. For each nonempty Π0 2 set X ⊆ [0, 1], in particular for X = {x}, DX is Π0 3 -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to… CONTINUE READING