Generalized arithmetic progression

In mathematics, a multiple arithmetic progression, generalized arithmetic progression, k-dimensional arithmetic progression or a linear set, is a set of integers or tuples 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 x in N n of the form

with c 0 in C , x 1 , … , x m in P , and k 1 , … , 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. The semilinear sets are exactly the sets definable in Presburger arithmetic.