Signed-bit representations of real numbers

Lubarsky, Robert S.; Richman, Fred

doi:10.4115/jla.2009.1.10

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…Specifically, we use the representation described in Loeb [12], which is slightly different from the one described in [16]. This representation is essentially the signed-digit representation of the unit interval, where each real number in the unit interval is represented by a path in the ternary tree (see e.g., Lubarsky and Richman [13]).…”

Section: Real-valued Functions On the Cantor Spacementioning

confidence: 99%

Decidable fan theorem and uniform continuity theorem with continuous moduli

Fujiwara

Kawai

2019

Preprint

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The uniform continuity theorem (UCT) states that every pointwise continuous real-valued function on the unit interval is uniformly continuous. In constructive reverse mathematics, UCT is stronger than the decidable fan theorem. In this paper, we show that when "pointwise continuous" is replaced with "having a continuous modulus", UCT becomes equivalent to the decidable fan theorem. Here, a modulus of a real-valued function on the unit interval is a function which calculates a modulus of pointwise continuity of the given function from the input and the required accuracy of the output. Such a modulus is said to be continuous if it is pointwise continuous with respect to the representation of real numbers as regular sequences of rationals equipped with the topology of Baire space. We also show that continuous real-valued functions on the unit interval which have continuous moduli are exactly those functions induced by type one "continuous functions" described by Loeb [Ann.

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Section: Real-valued Functions On the Cantor Spacementioning

confidence: 99%

Decidable fan theorem and uniform continuity theorem with continuous moduli

Fujiwara

Kawai

2019

Preprint

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“…Without CC, sequences are not the proper objects to represent real numbers. Another natural approach is through Dedekind cuts which is only applicable to ordered structures (see [4,5] for a comparison of different approaches in the context of constructive mathematics). Richman [8] replaced the notion of a metric space (which assumes the existence of R) by a structure he called a premetric space which only needs the rational numbers and presented a method for completing a premetric space.…”

Section: Introductionmentioning

confidence: 99%