1 research outputs found
Besicovitch-Eggleston sets for finite GLS number systems with redundancy
In this article we study Besicovitch-Eggleston sets for finite GLS number
systems with redundancy. These number systems produce number expansions
reminiscent of Cantor base expansions. The redundancy refers to the fact that
each number has uncountably many representations in the system.
We distinguish between these representations by adding an extra dimension and
describing the system as a diagonally affine IFS on . For the
associated two dimensional level sets of digit frequencies we give the Birkhoff
spectrum and an expression for the Hausdorff dimension. To obtain these results
we first prove a more general result on the Hausdorff dimension of level sets
for Birkhoff averages of continuous potentials for a certain family of
diagonally affine IFS's. We also study the Hausdorff dimension of digit
frequency sets along fibres.Comment: 21 pages, 1 figur