We define a flat dyadic block transform for subsets of the positive integers and forarbitrary positive sequences. The construction begins with the indicator function of a seton the number line, partitions the line into consecutive blocks of length 2w, and reads eachblock as a finite binary word, hence as an integer symbol. This produces, for each scalew ≥ 1, an exact symbolic sequence associated with the original set.Motivated by the multiscale representation of integer sets introduced by Melkemi 1,we isolate and study the flat, non-hierarchical dyadic layer as an object in its own right.We prove exact recoverability for sets, and we extend the construction to arbitrary positivesequences by means of a cumulative lifting that is strictly increasing and reversible by firstdifferences. We also define a sparse-support factorization of the symbolic sequence intozero-runs and nonzero symbols, which is natural for highly sparse sets.The point of view is arithmetical rather than purely computational: the transformassociates to each set or positive sequence a family of local symbolic signatures indexed bythe dyadic block size. Computational consequences such as effective alphabets, zero density,symbolic entropy, and sparse-support encodings appear as natural by-products of the theory.
Ricardo Adonis Caraccioli Abrego (Fri,) studied this question.