346 | Erdős Problems

Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that

$A\backslash B$ is complete for any finite subset $B$ and
$A\backslash B$ is not complete for any infinite subset $B$.

(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)

Is it true that if $a_{n+1}/a_n \geq 1+\epsilon$ for some $\epsilon>0$ and all $n$ then\[\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?\]

#346: [ErGr80,p.57]

number theory | complete sequences

Graham [Gr64d] has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erdős and Graham [ErGr80] remark that it is easy to see that if $a_{n+1}/a_n>\frac{1+\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.

Proof expositions (0)

If you would like to contribute an exposition of a proof related to this problem, please message a moderator or leave your exposition as a comment.

No proof expositions yet.

Comments (6) Proof claims (1)