What is it about?
This work explores a fundamental question: Does the density of information in a data stream remain unchanged across different representations? We investigate the continued fraction representation of data and show that the density of information can be at times significantly larger when using the continued fraction expansion. Density of information is a fundamental property of the data. This work shows that the density of information of a data stream can change according to the method of representation used.
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Why is it important?
This is the first setting under which non preservation of information density is demonstrated. This is interesting as the notion of randomness is unchanged under different representations. But this work says that for non random objects, the density of randomness does depend on the representation of the data in consideration.
Perspectives
This opens up research into more questions. 1. What are the nature of representations that preserve density of information. 2. Does it have any association with preservation of fractal dimension. 3. Are there other concrete examples of representations in which density of information is not preserved ?
Akhil S
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This page is a summary of: Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals, ACM Transactions on Computation Theory, March 2025, ACM (Association for Computing Machinery),
DOI: 10.1145/3723323.
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