On an alternative Cauchy equation (Q1070128)

Functions \(f: G\to {\mathbb{Z}}\) such that \(f(x+y)-f(x)-f(y)\in \{0,1,\ldots,k\}\) for all x,y\(\in G\) are described in the case where G is an Abelian group or the semigroup of all positive integers. It is also shown that some more general problems can be reduced to these ones.