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Source. Part 3 (Open problems), Problem 3, p. 8, of P. Erdős and M. B. Nathanson, "Partitions of bases into disjoint unions of bases," J. Number Theory 29 (1988), no. 1, 1--9. The edition read is identified on the source card.

Statement

Definition (p. 8). An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$.

Problem 3 (p. 8). The paper recalls that Härtter and Nathanson proved that there are asymptotic bases containing no minimal asymptotic basis, and that Erdős and Nathanson ("Systems of distinct representatives and minimal bases in additive number theory," Number Theory, Carbondale 1979, Lecture Notes in Math. 751, Springer, 1979, 89--107) proved: if $A$ is an asymptotic basis of order 2 with $f(n)\ge c\log n$ for some $c>\log^{-1}(4/3)$ and all $n\ge n_0$, then $A$ contains a minimal asymptotic basis of order 2. Here $f(n)$ is the count of Problem 2. The paper adds that the proof is similar to that of Theorem 1 but seems to work only in the case $h=2$.

It then states as unknown whether an asymptotic basis $A$ of order $h>2$ for which $f(n)\ge c\log n$, for some sufficiently large constant $c$, must contain a minimal asymptotic basis of order $h$. The problem does not restate $f(n)$ for $h>2$; for that order the paper's abstract and Theorems 4 and 5 use the size of a maximal family of pairwise disjoint representations of $n$ as a sum of $h$ elements.

It also records an older problem of Erdős and Nathanson from the 1979 paper: if $A$ is an asymptotic basis of order $h$ with $\lim_{n\to\infty}f(n)=\infty$, does $A$ contain a minimal asymptotic basis of order $h$? The paper says this is open even for $h=2$.

Read depth. Claims checked: the problem was read clause by clause on the print. The cited results of Härtter, Nathanson, and Erdős and Nathanson were not checked.

Dependencies

None.

Bears on

• #870: the problem page cites this paper as its source for this question. The site asks it for $k\ge3$ with $r(n)$ counting representations of $n$ as a sum of at most $k$ elements, where the paper asks it for $h>2$ with $f(n)$ as above. The paper poses the question and does not answer it.