Dense sets of integers with prescribed representation functions

Let A be a set of integers and let h ≥ 2 . For every integer n , let r A , h ( n ) denote the number of representations of n of the form n = a 1 + ⋯ + a h , where a i ∈ A for 1 ≤ i ≤ h , and a 1 ≤ ⋯ ≤ a h . The function r A , h : Z → N , where N = N ∪ { 0 , ∞ } , is the representation function of order h for A . ve that, given a positive integer g , every function f : Z → N satisfying lim inf ∣ n ∣ → ∞ f ( n ) ≥ g is the representation function of order h of a sequence A of integers “almost” as dense as any given B h [ g ] sequence. Specifically we prove that, given an integer h ≥ 2 and ε > 0 , there exists g = g ( h , ϵ ) such that for any function f : Z → N satisfying lim inf ∣ n ∣ → ∞ f ( n ) ≥ g there exists a sequence A satisfying r A , h = f and | A ∩ [ 1 , x ] | ≫ x ( 1 / h ) − ε . y speaking we prove that the problem of finding a dense set of integers with a prescribed representation function f of order h and lim inf ∣ n ∣ → ∞ f ( n ) ≥ g is “equivalent” to the classical problem of finding dense B h [ g ] sequences of positive integers.