Finite-State Dimension and Real Arithmetic
<|reference_start|>Finite-State Dimension and Real Arithmetic: We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.<|reference_end|>
@article{doty2006finite-state,
    title={Finite-State Dimension and Real Arithmetic},
    author={David Doty, Jack H. Lutz, Satyadev Nandakumar},
    journal={arXiv preprint arXiv:cs/0602032},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0602032},
    primaryClass={cs.CC cs.IT math.IT}
}