Sharpness of the one-dimensional Abelian maximal pattern-complexity lower bound

Determine whether the lower bound p_x^{*\mathrm{ab}}(k)\geq (r-1)k+1 is sharp for recurrent one-dimensional infinite words over an alphabet of size r that are aperiodic by projection; equivalently, determine whether equality can be attained in the one-dimensional setting.

Background

The paper studies Abelian maximal pattern complexity, which counts the distinct Parikh vectors of patterns of a fixed cardinality and then maximizes over all patterns. A previously known one-dimensional result gives the lower bound p_x^{*\mathrm{ab}}(k)\geq (r-1)k+1 for recurrent words that are aperiodic by projection over an alphabet of size r. The unresolved issue is whether this lower bound is always attainable, or otherwise genuinely sharp, in one dimension.

The paper contrasts this unresolved one-dimensional question with its two-dimensional results. In dimension two, Proposition 1.4 constructs, for every alphabet size r\geq 2, a word attaining equality p_x^{*\mathrm{ab}}(k)=(r-1)k+1 for every k\geq 1, thereby establishing sharpness in the two-dimensional setting.