SUPREMUM OF REPRESENTATION FUNCTIONS

摘要

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 N = N∪{∞}. 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.