Point-to-set Principle and Constructive Dimension Faithfulness
CoRR(2024)
摘要
We introduce a constructive analogue of Φ-dimension, a notion of
Hausdorff dimension developed using a restricted class of coverings of a set. A
class of coverings Φ is said to be "faithful" to Hausdorff dimension if
the Φ-dimension and Hausdorff dimension coincide for every set.
We prove a Point-to-Set Principle for Φ-dimension, through which we get
Point-to-Set Principles for Hausdorff Dimension, continued-fraction dimension
and dimension of Cantor Coverings as special cases. Using the Point-to-Set
Principle for Cantor coverings and a new technique for the construction of
sequences satisfying a certain Kolmogorov complexity condition, we show that
the notions of faithfulness of Cantor coverings at the Hausdorff and
constructive levels are equivalent.
We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin to derive the
necessary and sufficient conditions for the constructive dimension faithfulness
of the coverings generated by the Cantor series expansion. This condition
yields two general classes of representations of reals, one whose constructive
dimensions that are equivalent to the constructive Hausdorff dimensions, and
another, whose effective dimensions are different from the effective Hausdorff
dimensions, completely classifying Cantor series expansions of reals.
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