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
where and so on are fixed, and and so on are confined to some ranges
- ≤ ≤
and so on, for a finite progression. The number , 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.