Sparse subset of $\mathbb{N}$ with a summation property For $A\subseteq \mathbb{N}$ and an integer $k\geq 1$ we set $S_A(k) = \{B\subseteq A: B\text{ is finite and } \sum_{b\in B} b = k\}.$
We say that a set $A\subseteq \mathbb{N}$ is sparse if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$
Is there a sparse set $A\subseteq \mathbb{N}$ such that for every $n\in\mathbb{N}$ there is an integer $z\in \mathbb{N}$ such that $$|S_A(z)| \geq n?$$
 A: Summarizing the commentary of Will Brian and myself, there is a notion of IP set (https://en.m.wikipedia.org/wiki/IP_set) (edit: Will says "has your summation property", which I misinterpreted to mean the following ) which has the property that Dominic points out: for every positive integer $n$ there is a member (in fact an infinite subset of them) $z_n$ of the IP set which is the sum in at least $n$ different ways of the members of the base set $A$ which generates the IP set.
Dominic wants a thin or sparse set $A$ to build an IP set. The Fibonacci sequence F provides an example of a sparse $A$ with Dominic's property, as every sufficiently large member of F itself has multiple representations as sums of Fibonacci numbers.  However F is not an IP set ( as an infinite IP set has a partial closure property: given a in the IP set there are infinitely many b in the IP set with a+b also in the IP set). Will mentions the example of base 10 numbers having only ones or zeros in their decimal expansion as a sparse IP set.
One can make A, S(A), and further iterations of S on A simultaneously sparse with a modification of the following technique: let A have 1, and for each positive integer n, throw in c=100^{100^n} and c-a for all a put in at an earlier stage. Then A and S(A) have small but widely separated dense clumps of integers, and one can arrange further iterates to be sparse.
Gerhard "And That Ends The Commentary" Paseman, 2017.04.24.
