Some subsets of set ℱ in the diluted Hofstadter problem

We continue work started in Deane and Gentile [A diluted version of the problem of the existence of the Hofstadter sequence, J. Differ. Equ. Appl. 31 (2024), pp. 48–65] concerning integer sequences (Formula presented.), (Formula presented.), defined by (Formula presented.), with (Formula presented.). Here, (Formula presented.), with (Formula presented.), is a given sequence. We define (Formula presented.) as the set of semi-infinite sequences f such that the resulting sequence q exists. This requires that the term (Formula presented.) be defined for all n, that is, (Formula presented.) applies for all (Formula presented.). We use a variety of approaches to probe the structure of (Formula presented.), including explicit construction, analysis and computer-assisted proof.