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- W2024776286 abstract "Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano (1993) and Bugeaud (2002) have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k ≥ 2, we give a new construction of a Liouville number which is normal to the base k. This construction is combinatorial, and is based on de Bruijn sequences. A real number in the unit interval is normal if and only if its finite-state dimension is 1. We generalize our construction to prove that for any rational r in the closed unit interval, there is a Liouville number with finite state dimension r. This refines Staiger’s result [19] that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state dimension." @default.
- W2024776286 created "2016-06-24" @default.
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- W2024776286 date "2014-06-08" @default.
- W2024776286 modified "2023-09-27" @default.
- W2024776286 title "Normality and Finite-State Dimension of Liouville Numbers" @default.
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- W2024776286 doi "https://doi.org/10.1007/s00224-014-9554-8" @default.
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