Effective Versions of Strong Measure Zero

TL;DR

This paper develops effective versions of strong measure zero sets across various complexity levels, introduces new characterizations, and explores their properties in computability and algorithmic contexts.

Contribution

It introduces effective definitions of strong measure zero using odds supermartingales and provides a detailed analysis across classical, bounded, and computable frameworks.

Findings

Borel's conjecture holds in the bounded setting but not in the computable setting.

The paper characterizes computable strong measure zero sets within the hyperarithmetical hierarchy.

An algorithmic version of strong measure zero is shown to be equivalent to NCR reals.

Abstract

Effective versions of strong measure zero sets are developed for various levels of complexity and computability. It is shown that the sets can be equivalently defined using a generalization of supermartingales called odds supermartingales, success rates on supermartingales, predictors, and coverings. We show Borel's conjecture of a set having strong measure zero if and only if it is countable holds in the time and space bounded setting. At the level of computability this does not hold. We show the computable level contains sequences at arbitrary levels of the hyperarithmetical hierarchy by proving a correspondence principle yielding a condition for the sets of computable strong measure zero to agree with the classical sets of strong measure zero. An algorithmic version of strong measure zero using lower semicomputability is defined. We show that this notion is equivalent to the set of NCR reals studied by Reimann and Slaman, thereby giving new characterizations of this set. Effective strong packing dimension zero is investigated requiring success with respect to the limit inferior instead of the limit superior. It is proven that every sequence in the corresponding algorithmic class is decidable. At the level of computability, the sets coincide with a notion of weak countability that we define.