Convergence rates for computably selected bits

Determine whether the conditional probability error for Solomonoff induction decreases rapidly to zero on the bits fixed by a computable pattern, and, in particular, whether the sum of the errors remains bounded when every even bit equals the preceding odd bit.

Background

The paper studies prediction errors only on positions selected or constrained by a computable pattern, rather than on every bit of the sequence. Although prior work established convergence to zero for such selected bits, quantitative convergence bounds were unavailable. The authors reproduce a question previously raised in the literature asking whether rapid convergence and bounded cumulative error hold in this setting.

The paper subsequently proves upper bounds for powers greater than one and shows that the absolute-error sum can diverge for a suitable computable pattern. Thus, the broad question of rapid convergence is addressed only partially, while the specific boundedness behavior for the absolute loss remains unresolved in the form posed here.