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.