Generalized arithmetic progression

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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

a + mb + nc + \ldots

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


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.

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