On Erdős and Graham's \(X\) function. (Q1774077)

Let \(A\) be an asymptotic basis for the positive integers. \textit{P. Erdős} and \textit{R. L. Graham} [Acta Arith. 37, 201--207 (1980; Zbl 0443.10036)] proved that for all but finitely many \(a\in A\) the set \(A \setminus \{a\}\) is also a basis, and its order can be bounded by a quantity \(X(h)\) depending only on the order \(h\) of the basis \(A\). They also proved that \(X(h)/h^2\) stays between positive bounds. These bounds were improved later by G. Grekos and J. C. M. Nash. Here these bounds are further improved slightly; the new bounds are \[ [h(h+4)/3] \leq X(h) \leq h(h+1)/2 + [(h+1)/3] . \] As a byproduct of the methods used the author proves the following result. Let \(E\) be a set of residues modulo \(n\) such that every residue is a sum of at most \(h\) elements of \(E\). Then the set \(h'E\) is a union of certain nontrivial residue classes for \(h' = h(h+1)/2 + [(h+1)/3] \).