I would like to understand the conditions that support
the [Green-Tao Theorem](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem), which established that
the primes contain arbitrarily long arithmetic progressions.


I am wondering:



> 
> ***Q***. Is it difficult to define an infinite
>  set $S$ of natural numbers that mimics the 
>  $\sim n / \log n$ distribution of
>  the primes (and perhaps other prime properties), but yet fails to contain arbitrarily long arithmetic progressions?
> 
> 
> 


For example, $S$ might be the
[lucky numbers](https://en.wikipedia.org/wiki/Lucky_number)
([OEIS A000959](https://oeis.org/A000959)),
or
[Hawkins' "random primes."](https://www.jstor.org/stable/3029322?seq=1#metadata_info_tab_contents)
Do either of these avoid arbitrarily long arithmetic progressions?


My intuition is that it is easier to avoid such progressions
than it is to establish their existence.




---


MSE question unanswered:
[Do lucky numbers contain arbitrarily long arithmetic progressions?](https://math.stackexchange.com/q/3660824/237)



Prime-like numbers that avoid Green-Tao?
This is likely impossible. 


Indeed the largest sets known to be free of arbitrarily long arithmetic progressions asymptotically satisfy $| [ A \cap [1, n] | \lesssim\_{k} n / \log^{k} n$ for all $k>1$, and it is widely believed that these examples are near maximal.


The Erdos-Turan conjecture mentioned in the comments is almost equivalent to the claim that any set of relative density greater than $n/ \log n$ contains arbitrary long arithmetic progressions. As remarked above, this is likely true for even sparser sets.


