Dense sets of integers with prescribed representation functions

TL;DR

This paper demonstrates that for any sufficiently large representation function, there exists a set of integers with a density close to x^{1/h} that realizes this function, extending the understanding of additive representation structures.

Contribution

It constructs integer sets with prescribed representation functions that grow nearly as slowly as any given B_h[g] sequence, generalizing previous results.

Findings

Existence of sets with prescribed representation functions for large n

Construction of sets with near minimal growth rate

Extension of additive representation theory

Abstract

Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is called the representation function of order h for the set A. We prove that every function f from Z to N_0 U {\infty} satisfying liminf_{|n|->\infty} f (n)\geq g is the representation function of order h for some sequence A of integers, and that A can be constructed so that it increases "almost" as slowly as any given B_h[g] sequence. In particular, for every epsilon >0 and g \geq g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\gg x^{(1/h)-epsilon}.