k-spectra of weakly-c-balanced words

A word u is a scattered factor of w if u can be obtained from w by deleting some of its letters. That is, there exist the (potentially empty) words u₁, u₂, ..., u_n, and v₀, v₁, .., v_n such that u = u₁u₂...u_n and w = v₀u₁v₁u₂v₂...u_nv_n. We consider the set of length-k scattered factors of a given word w, called here k-spectrum and denoted ScatFact_k (w). We prove a series of properties of the sets ScatFact_k (w) for binary weakly-0-balanced and, respectively, weakly-c-balanced words w, i.e., words over a two-letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has c-more occurrences than the other. In particular, we consider the question which cardinalities n = | ScatFact_k (w)| are obtainable, for a positive integer k, when w is either a weakly-0-balanced binary word of length 2k, or a weakly-c-balanced binary word of length 2k − c. We also consider the problem of reconstructing words from their k-spectra.