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 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. The semilinear sets are exactly the sets definable in Presburger arithmetic.[1]
See also
References
- ↑ Ginsburg, Seymour; Spanier, Edwin Henry (1966). "Semigroups, Presburger Formulas, and Languages". Pacific Journal of Mathematics. 16: 285–296.
- Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and Geometry of Sumsets. Graduate Texts in Mathematics. 165. Springer. ISBN 0-387-94655-1. Zbl 0859.11003.