Dense sets of integers with prescribed representation functions

Let A be a set of integers and let h≥2. For every integer n, let rA,h(n) denote the number of representations of n of the form n=a1+⋯+ah, where ai∈A for 1≤i≤h, and a1≤⋯≤ah. The function rA,h:Z→N, where N=N∪{0,∞}, is the representation function of order hfor A.We prove 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 Bh[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 rA,h=f and |A∩[1,x]|≫x(1/h)−ε.Roughly 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 Bh[g] sequences of positive integers.