Nova S¶erie EVERY FUNCTION IS THE REPRESENTATION FUNCTION OF AN ADDITIVE BASIS FOR THE INTEGERS

Abstract: Let A be a set of integers. For every integer n, let rA;h(n) denote the number of representations of n in the form n = a1+a2+ ¢ ¢ ¢+ah; where a1; a2;:::; ah 2 A and a1 · a2 · ¢ ¢ ¢ · ah: The function rA;h: Z! N0 [ f1g is the representation function of order h for A. The set A is called an asymptotic basis of order h if r¡1A;h(0) is ¯nite, that is, if every integer with at most a ¯nite number of excep-tions can be represented as the sum of exactly h not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z! N0[f1g is any function such that f¡1(0) is ¯nite, then there exists a set A of integers such that f(n) = rA;h(n) for all n 2 Z. Moreover, the set A can be arbitrarily sparse in the sense that, if '(x) ¸ 0 for x ¸ 0 and '(x)!1, then there exists a set A with f(n) = rA;h(n) and card (fa 2 A: jaj · xg) < '(x) for all x. It is an open problem to construct dense sets of integers with a prescribed represen-tation function. Other open problems are also discussed