A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension

2020/11/21 by Coons, Michael, Evans, James
#52C23 #FOS: Mathematics #Number Theory (math.NT) #Primary 28A80 Secondary 11K55

paper · doi:10.48550/arxiv.2011.10722

Abstract

We show that missing q-ary digit sets F⊆[0,1] have 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 ν_F supported on F is equal to the ghost measure μ_f of f.

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