On large half-factorial sets in elementary \(p\)-groups: maximal cardinality and structural char\-ac\-ter\-i\-zation. (Q1775477)

With a subset \(G_0\) of a finite Abelian group \(G\) (written additively) one associates the free monoid \({\mathcal F}(G_0)\) whose free generators are elements of \(G_0\), and the block monoid \({\mathcal B}(G_0)\), consisting of all \(\prod_{i=1}^mg_i\in{\mathcal F}(G_0)\) with \(\sum_{i=1}^mg_i=0\). \(G_0\) is called half-factorial, if \({\mathcal B}(G_0)\) is half-factorial. The authors obtain an explicit formula for \(\mu(G)\), the maximal cardinality of a half-factorial subset of \(G\) in the case when \(G=C_p^n\). This extends a result of \textit{A. Geroldinger} and \textit{J. Ka\-czo\-rowski} [Sémin. Théor. Nombres, Bordx., Sér. II 4, No. 2, 199-238 (1992; Zbl 0780.11046)], who obtained this formula in the case of even \(n\). Moreover, the authors determine the structure of half-factorial subsets \(G_0\) of \(C_p^n\) having at least \(\mu(G)-cp\) elements, with \(c\) being an absolute constant. These results are then applied to the problem of oscillations of the error term in the counting formula for the set of non-associated integers in an algebraic number field (with class-group \(C_p^n\)) having factorizations of at most \(k\) different lengths.