Problem
Research questions and friction points this paper is trying to address.
Relativizes normality in number representation systems.
Introduces supernormal and highly normal numbers.
Links high normality to computable dimension 1 sequences.
Normality, Relativization, and Randomness
🤖 AI Summary
This paper addresses the relativization of Borel absolute normality within generalized digital representation systems, establishing a natural relativization framework for genericity. It introduces two novel notions of algorithmic randomness: “hyper-normal numbers” and “high-normal numbers.” The former strictly lies between 2-randomness and effective dimension one; the latter provides the first purely generic characterization of sequences with computable dimension one. Combining techniques from algorithmic information theory, computability theory, and generalized base analysis, the authors prove both notions constitute proper generalizations of Borel absolute normality and precisely locate them within the algorithmic randomness hierarchy. Key contributions include: (i) the first relativized model of genericity; (ii) an intrinsic generic characterization of computable dimension-one sequences; and (iii) a deeper conceptual unification of number-theoretic normality and algorithmic randomness.
📝 Abstract
Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of supernormal numbers that correspond to semicomputable relativizations, and that of highly normal numbers in terms of computable ones. We prove several properties of these new randomness concepts. Both supernormality and high normality generalize Borel absolute normality. Supernormality is strictly between 2-randomness and effective dimension 1, while high normality corresponds exactly to sequences of computable dimension 1 providing a more natural characterization of this class.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Relativization of normality
Supernormal numbers concept
High normality characterization
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