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:

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?)