Real numbers equally compressible in every base
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Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence. Absolutely normal numbers, being finite-state incompressible in every base of expansion, have finite-state dimension equal to 1 in every base-b. At the other extreme, every rational number has finite-state dimension equal to 0 in every base b. Generalizing this, Lutz and Mayordomo posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number ξ and a compressibility ratio s ∈ (0,1) such that for every base b, the compressibility ratio of the base-b expansion of ξ is precisely s? In this paper, we answer this affirmatively by proving a more general result. We show that given any sequence of rational dimensions (compressibility ratios) ⟨ q_b ⟩_b=1^∞ in natural number bases, we can explicitly construct a single number ξ such that for any base b, the finite-state dimension, or equivalently, compression ratio, of ξ in base-b is q_b. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between 0 and 1, as posed by Lutz and Mayordomo. In our construction, we combine ideas from Wolfgang Schmidt's construction of absolutely normal numbers (1962), results regarding low discrepancy sequences and several new estimates related to exponential sums.
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