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Extending the Limit Theorem of Barmpalias and Lewis-Pye to all reals

Ivan Titov

TL;DR

The paper extends the Limit Theorem of Barmpalias and Lewis-Pye from left-c.e. Martin-Löf random reals to all reals by formulating the result in terms of nondecreasing translation functions between reals. It shows that for any real $\alpha$ and Martin-Löf random real $\beta$, there exists a constant $d\ge0$ such that for every nondecreasing translation function $g$ from $\beta$ to $\alpha$, the left-hand limit $\lim_{q\nearrow\beta} (\alpha - g(q))/(\beta - q)$ exists and equals $d$, with $d=0$ iff $\alpha$ is not Martin-Löf random. The work decomposes the proof into three core parts: bounding the fraction via a Solovay-reducibility argument, proving the existence of the left limit, and establishing its independence from the chosen translation function, thereby connecting index and rational formulations. It also discusses extensions to broader reducibility notions (notably S2a) and potential implications for Schnorr randomness, outlining conjectures and directions for future research. $

Abstract

By a celebrated result of Kučera and Slaman (DOI:10.1137/S0097539799357441), the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye (arXiv:1604.00216) strengthened this result by showing that, for all left-c.e. reals $α$ and $β$ such that $β$ is Martin-Löf random and all left-c.e. approximations $a_0,a_1,\dots$ and $b_0,b_1,\dots$ of $α$ and $β$, respectively, the limit \begin{equation*} \lim\limits_{n\to\infty}\frac{α- a_n}{β- b_n} \end{equation*} exists and does not depend on the choice of the left-c.e. approximations to $α$ and $β$. Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.

Extending the Limit Theorem of Barmpalias and Lewis-Pye to all reals

TL;DR

The paper extends the Limit Theorem of Barmpalias and Lewis-Pye from left-c.e. Martin-Löf random reals to all reals by formulating the result in terms of nondecreasing translation functions between reals. It shows that for any real and Martin-Löf random real , there exists a constant such that for every nondecreasing translation function from to , the left-hand limit exists and equals , with iff is not Martin-Löf random. The work decomposes the proof into three core parts: bounding the fraction via a Solovay-reducibility argument, proving the existence of the left limit, and establishing its independence from the chosen translation function, thereby connecting index and rational formulations. It also discusses extensions to broader reducibility notions (notably S2a) and potential implications for Schnorr randomness, outlining conjectures and directions for future research. $

Abstract

By a celebrated result of Kučera and Slaman (DOI:10.1137/S0097539799357441), the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye (arXiv:1604.00216) strengthened this result by showing that, for all left-c.e. reals and such that is Martin-Löf random and all left-c.e. approximations and of and , respectively, the limit \begin{equation*} \lim\limits_{n\to\infty}\frac{α- a_n}{β- b_n} \end{equation*} exists and does not depend on the choice of the left-c.e. approximations to and . Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.
Paper Structure (6 sections, 10 theorems, 169 equations)

This paper contains 6 sections, 10 theorems, 169 equations.

Table of Contents

  1. Introduction and background
  2. The theorem
  3. The fraction is bounded from above
  4. The left limit exists
  5. The left limit is unique
  6. Conclusion and further extensions

Key Result

Proposition 1.4

A left-c.e. real $\alpha$ is Solovay reducible to a left-c.e. real $\beta$ with a Solovay constant $c$ if and only if, for every left-c.e. approximations $a_0,a_1,\dots\nearrow\alpha$ and $b_0,b_1,\dots\nearrow\beta$, there exists a computable index function $f:\mathbb{N}\to\mathbb{N}$ such that, fo

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: Solovay, 1975
  • Proposition 1.4: Calude et al., 1998
  • Proposition 1.5
  • Remark
  • Lemma 1.6: Kučera and Slaman, 2001; explicitly: Miller, 2017
  • Theorem 1.7: Barmpalias, Lewis-Pye, 2017
  • Theorem 1.8: Rational form of the Limit Theorem of Barmpalias and Lewis-Pye
  • Proposition 1.9
  • ...and 49 more