A mapping is called residue class-wise affine if there is a nonzero integer such that the restrictions of to the residue classes (mod ) are all affine. This means that for any residue class there are coefficients such that the restriction of the mapping to the set is given by
Residue class-wise affine groups are countable, and they are accessible to computational investigations. 'Many' of them act multiply transitively on or on subsets thereof. Only relatively basic facts about their structure are known so far.
See also the Collatz conjecture, which is an assertion about a surjective, but not injective residue class-wise affine mapping.