Additive bases in groups (Q522315)

Let \(G\) be a commutative semigroup written additively. A subset \(A\) of \(G\) is said to be a basis of order at most \(h\) of \(G\) if all but finitely many elements of \(G\) can be written as a sum of exactly \(h\) elements of \(A.\) If \(h\) is the smallest integer for which this holds, one says that \(A\) is a basis of order \(h\) of \(G,\) and write \(\mathrm{ord}^{*}_{G}(A)= h.\) If no such \(h\) exists, then \(\mathrm{ord}^{*}_{G}(A)= \infty.\) A basis \(A\) of order \(h\) of \(G\) is minimal if for any \(a \in A,\) \(A \setminus \{a\}\) is no longer a basis of order \(h,\) and the basis \(A\) is nice if \(hA = G.\) In the paper under review, the authors prove Theorem 1. Let \(G\) be any infinite abelian group and \(h\) be an integer \(\geq 2.\) Then \(G\) has a nice minimal basis of order \(h.\) Given a basis \(A\) of an abelian group \(G,\) an element \(a \in A\) is called exceptional if \(G \setminus \{a\}\) is not a basis of \(G\) of any order, and regular if \(a\) is not exceptional. Letting \(A^{*}\) denote the set of regular elements in \(A,\) the authors define \[ E_{G}(h) = \max_{hA \sim G}| A \setminus A^{*} | \] and \[ X_{G}(h) = \max_{hA \sim G} \max_{a \in A^{*}}\mathrm{ord}^{*}_{G}(A \setminus A^{*}), \] where \(hA \sim G\) means that the symmetric difference of \(hA\) and \(G\) is finite. The authors establish some results about the functions \(E_{G}\) and \(X_{G}.\) For instance, they prove: Theorem 2. \(\;\) (i) For any infinite abelian group \(G\) and any integer \(h \geq 2,\) we have \(E_{G}(h) \leq h - 1.\) (ii) There is an infinite group \(G\) for which \(E_{G}(h) = h\) for any integer \(h \geq 2.\) (iii) For each integer \(h \geq 2,\) there is an infinite group \(G\) (depending on \(h\)) for which \(E_{G}(h) = 0.\) Theorem 3. Let \(G\) be an infinite abelian group. Suppose there is a subgroup \(H\) of \(G\) such that \(G/H \cong \mathbb{Z}.\) Then for any integer \(h \geq 1,\) we have \(X_{G}(h) \geq \Bigl[\frac{h(h+4)}{3}\Bigr].\)