An Erdős–Fuchs theorem for ordered representation functions

On $B_{h}[1]$-sets which are asymptotic bases of order $2h$

“…In this paper we continue the work in this direction. Particularly, we improve on the above result by proving the existence of a B h [1]-set which is an asymptotic basis of order 2h.…”

“…can be formulated in a more general form. In a recent paper [11], we proved the existence of a B h [1]-set which is at the same time forms an asymptotic basis of order 2h + 1.…”

“…where n ∈ N. We say a set of positive integers A forms a B h [g]-set if for every n ∈ N, the number of representations of n as the sum of h terms in the form (1) is at most g, that is R h,A (n) ≤ g. A set A ⊂ N is said to be an asymptotic basis of order k if there exists a positive integer n 0 such that R k,A (n) > 0 for n > n 0 . In [4] and [5], P. Erdős, A. Sárközy and V. T. Sós asked if there exists a Sidon set (i.e., a B 2 [1]-set) which is an asymptotic basis of order 3. It is easy to see that a Sidon set cannot be an asymptotic basis of order 2 because it does not have enough elements.…”

“…Theorem 1. For every h ≥ 2 integer there exists an asymptotic basis of order 2h which is a B h [1]-set.…”

“…These problems are also appear in [11]. In general, for k, h ≥ 2 integers, one can be interested in the existence of an asymptotic basis of order k which is a B h [1]-set. It is easy to see that there does not exist an asymptotic basis of order k < h which is a B h [1]-set because it does not have enough elements.…”