Journal of Integer Sequences, Vol. 24 (2021), Article 21.2.5

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.

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(Concerned with sequences A088917 A292686.)

Received October 18 2020; revised version received January 28 2021. Published in Journal of Integer Sequences, January 28 2021.

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