Generalized arithmetic progression

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

a + mb + nc + ...

where a, b, c and so on are fixed, and m, n and so on are confined to some ranges

0 ≤ m ≤ M,

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

$L(C;P)$

be the set of all elements $x$ in $N^{n}$ of the form

$x=c_{0}+\sum _{i=1}^{m}k_{i}x_{i}$,

with $c_{0}$ in $C$, $x_{1},\ldots ,x_{m}$ in $P$, and $k_{1},\ldots ,k_{m}$ in $N$. $L$ is said to be a linear set if $C$ consists of exactly one element, and $P$ is finite.

A subset of $N_{n}$ is said to be semilinear if it is a finite union of linear sets.