In mathematics, a multiple arithmetic progression, generalized arithmetic progression, or k-dimensional arithmetic progression, is a set of integers constructed as an arithmetic progression is, but allowing several possible differences. So, for example, we start at 17 and may add a multiple of 3 or of 5, repeatedly. In algebraic terms we look at integers
where a, b, c and so on are fixed, and m, n and so on are confined to some ranges
and so on, for a finite progression. The number k, that is the number of permissible differences, is called the dimension of the generalized progression.
More generally, let
be the set of all elements in of the form
with in , in , and in . is said to be a linear set if consists of exactly one element, and is finite.
A subset of is said to be semilinear if it is a finite union of linear sets.