Birkhoff spectrum for diagonally self-affine sets and digit frequencies for GLS systems with redundancy

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine IFS’s. Also, we study Besicovitch-Eggleston sets for finite GLS number systems with redundancy. The redundancy refers to the fact that each number $x\in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.