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Supremum of Representation Functions

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Published/Copyright: August 7, 2011
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Integers
From the journal Volume 11 Issue 5

Abstract

For a subset A of ℕ = {0, 1, 2, . . .}, the representation function of A is defined by rA(n) = |{(a, b) ∈ A × A : a + b = n}|, for n ∈ ℕ, where |E| denotes the cardinality of a set E. Its supremum is the element s(A) = sup{rA(n) : n ∈ ℕ} of . Interested in the question “when is s(A) = ∞?”, we study some properties of the function A ↦ s(A), determine its range, and construct some subsets A of ℕ for which s(A) satisfies certain prescribed conditions.

Keywords.: Representation Function; Bases; Caliber
Published Online: 2011-08-07
Published in Print: 2011-October

© de Gruyter 2011

Articles in the same Issue

  1. On Congruent Numbers with Three Prime Factors
  2. Projective p-Orderings and Homogeneous Integer-Valued Polynomials
  3. Analyzing Two-Color Babylon
  4. On the Number of Points in a Lattice Polytope
  5. Combinatorial Proofs of Some Identities for the Fibonacci and Lucas Numbers
  6. An Extreme Family of Generalized Frobenius Numbers
  7. On the Distance Between Smooth Numbers
  8. Analogs of the Stern Sequence
  9. On Some Equations Related to Ma's Conjecture
  10. Normality, Projective Normality and EGZ Theorem
  11. The (Exponential) Bipartitional Polynomials and Polynomial Sequences of Trinomial Type: Part II
  12. Supremum of Representation Functions
https://doi.org/10.1515/integ.2011.054
Keywords for this article
Representation Function; Bases; Caliber

Articles in the same Issue

  1. On Congruent Numbers with Three Prime Factors
  2. Projective p-Orderings and Homogeneous Integer-Valued Polynomials
  3. Analyzing Two-Color Babylon
  4. On the Number of Points in a Lattice Polytope
  5. Combinatorial Proofs of Some Identities for the Fibonacci and Lucas Numbers
  6. An Extreme Family of Generalized Frobenius Numbers
  7. On the Distance Between Smooth Numbers
  8. Analogs of the Stern Sequence
  9. On Some Equations Related to Ma's Conjecture
  10. Normality, Projective Normality and EGZ Theorem
  11. The (Exponential) Bipartitional Polynomials and Polynomial Sequences of Trinomial Type: Part II
  12. Supremum of Representation Functions
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