Lowness for Integer-Valued Randomness

TL;DR

This paper characterizes the concept of lowness for integer-valued randomness, showing it coincides with recursiveness, similar to the case for computable randomness, thus deepening understanding of randomness notions.

Contribution

It establishes that lowness for integer-valued randomness is equivalent to recursiveness, providing a precise characterization of this lowness property.

Findings

Lowness for integer-valued randomness coincides with recursiveness.

The result parallels the case for computable randomness.

Provides a clear criterion for lowness in this context.

Abstract

A real is called integer-valued random if no integer-valued martingale can win arbitrarily much capital betting against it. A real is low for integer-valued randomness if no integer-valued martingale recursive in A can succeed on an integer-valued random real. We show that lowness for integer-valued randomness coincides with recursiveness, as is the case for computable randomness.