Complementing sets of $n$-tuples of integers

Linear Forms and Complementing Sets of Integers

Let φ(x1, . . . , xh, y) = u1x1 + · · · + uhxh + vy be a linear form with nonzero integer coefficients u1, . . . , uh, v. Let A = (A1, . . . , Ah) be an h-tuple of finite sets of integers and let B be an infinite set of integers. Define the representation function associated to the form φ and the sets A and B as follows: R (φ) A,B (n) = card ({(a1, . . . , ah, b) ∈ A1 × · · · × Ah × B : φ(a1, ....

K-complementing Subsets of Nonnegative Integers

Let S = {S1,S2, . . .} represent a collection of nonempty sets of nonnegative integers in which each member contains the integer 0. Then S is called a complementing system of subsets for X ⊆ {0,1, . . .} if every x ∈ X can be uniquely represented as x = s1 + s2 + ··· with si ∈ Si. We will also write X = S1 ⊕ S2 ⊕ ··· and, when necessary, refer to X as the direct sum of the Si. We will denote th...

Tuples of Disjoint NP-Sets

Disjoint NP-pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2. We define subclasses of the class of all disjoint k-tuples of NP-sets. These subclasses are associated with a proposition...

N-tuples of Positive Integers with the Same Sum and the Same Product

It is proved that for every k there exist k triples of positive integers with the same sum and the same product. In this paper we solve the problem D.16 from the book [1] by proving the following Theorem. For every k there exist infinitely many primitive sets of k triples of positive integers with the same sum and the same product. (A set S of triples is called primitive if the greatest common ...

Periodic Sets of Integers