TITLE: Inverse Length 3 Arithmetic Progression Problem for sets with positive upper density
QUESTION [6 upvotes]: It is a famous theorem of Roth, which Szemerédi famously generalized, that if a set of natural numbers has positive upper density then it contains arithmetic progressions of length $k$. The famous Green-Tao Theorem generalized this property to the primes. My question is, is there any progress on the 'inverse' problem?
First Question: Suppose $A \subset \mathbb{N}$ has positive upper density. Does it follow that with at most finitely many exceptions, all elements $a \in A$ is in an arithmetic progression of length at least $3$ in elements of $a$? That is, with at most finitely many exceptions, is it true that for each $a \in A$ there exists $k > 0$ such that $a, a+k, a+2k \in A$?
Edit: This question has been answered; see below by two constructions. However a second question may be asked.
A related problem (which I believe to be harder) is the same question for the primes, which has been asked here before: 
Are all primes in a PAP-3?
Another related problem is found here (the constructions given all have density less than 1/2. Is it possible to find counterexamples with large density?)
Do there exist sets of integers with arbitrarily large upper density which contains infinitely many elements that are not in an arithmetic progression of length 3?

REPLY [11 votes]: Or just take all powers of $3$ and add to them all numbers that are congruent to $1$ modulo $3$.