Regainingly approximable numbers and sets

We call an $α\in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $α$ with $α- a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We also call a set $A\subseteq\mathbb{N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. $K$-trivial, we construct such an $α$ such that ${K(α\restriction n)>n}$ for infinitely many $n$. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

💡 Research Summary

The paper introduces a novel notion called “regainingly approximable” for real numbers and for subsets of the natural numbers. A real α is regainingly approximable if there exists a computable non‑decreasing sequence of rationals (aₙ) converging to α such that the inequality α − aₙ < 2⁻ⁿ holds for infinitely many indices n. This sits strictly between computable reals (which satisfy the inequality for all n) and left‑computable reals (which only require convergence of a non‑decreasing computable sequence). The authors first prove that the definition is robust: variations such as requiring a strictly increasing sequence, or replacing 2⁻ⁿ by 2⁻ᶠ⁽ⁿ⁾ for any computable unbounded function f, yield the same class (Proposition 4). Moreover, any left‑computable α that is regainingly approximable remains so regardless of which computable approximation is examined (Proposition 5).

The second part of the work focuses on sets A⊆ℕ for which the strongly left‑computable real 2⁻ᴬ = ∑_{a∈A}2^{-(a+1)} is regainingly approximable; such sets are called regainingly approximable sets. Using the standard characterization of left‑computable reals via uniformly computable left‑approximations (Lemma 7), the authors give several equivalent formulations (Theorem 8). In particular, a set A is regainingly approximable iff there exists a uniformly computable left‑approximation (Aₙ) such that A∩{0,…,n−1}=Aₙ for infinitely many n. This “infinitely often exact” property mirrors the real‑number definition.

The paper then investigates closure properties and degree‑theoretic aspects. It shows that the family of regainingly approximable sets is not closed under union or intersection, providing explicit counterexamples (Section 8). Despite this, every computably enumerable (c.e.) Turing degree contains a regainingly approximable set (Theorem 12), establishing the ubiquity of the notion across the degree structure. Conversely, there exist c.e. sets that are not regainingly approximable, confirming that the class is a proper subclass of the c.e. sets.

A significant portion of the study is devoted to Kolmogorov complexity. All regainingly approximable reals are “infinitely often K‑trivial”, hence they cannot be Martin‑Löf random. Nevertheless, the authors construct a regainingly approximable real α for which K(α↾n) > n for infinitely many n (Theorem 15), demonstrating that i.o. K‑triviality does not preclude occasional high complexity. Analogously, they exhibit regainingly approximable c.e. sets whose initial segment complexity reaches the maximal possible value for c.e. sets infinitely often (Theorem 16). These results highlight a nuanced relationship between the new approximation property and algorithmic information theory.

Finally, the paper addresses the splitting problem for regular reals (as defined by Wu). A regular real is a left‑computable number admitting a computable left‑approximation with a uniform speed bound. The authors present a uniform algorithm that, given any regular real, splits it into two reals that are both regular and regainingly approximable (Theorem 18). This strengthens earlier splitting results for c.e. sets and shows that the regainingly approximable property can be preserved under effective partitioning while maintaining regularity.

In summary, the work establishes regainingly approximable numbers and sets as a natural intermediate class between computable and left‑computable objects, explores their algebraic and degree‑theoretic behavior, connects them to Kolmogorov complexity, and provides effective splitting constructions. The results enrich our understanding of effective approximation, offering new tools for the study of algorithmic randomness, degree theory, and computable analysis.