A Sequential View of Self-Similar Measures; or, What the Ghosts of Mahler and Cantor Can Teach Us About Dimension
Michael Coons and James Evans
School of Mathematical and Physical Sciences
The University of Newcastle
Callaghan, NSW 2308
We show that a missing q-ary digit set F ⊆ [0,1] has a
corresponding naturally associated countable binary q-automatic
sequence f. Using this correspondence, we show that the Hausdorff
dimension of F is equal to the base-q logarithm of the
Mahler eigenvalue of f. In addition, we demonstrate that the
standard mass distribution νF
supported on F is equal to
the ghost measure μf of f.
Full version: pdf,
(Concerned with sequences
Received October 18 2020; revised version received January 28 2021.
Published in Journal of Integer Sequences,
January 28 2021.
Journal of Integer Sequences home page