While reading the statement of Roth's theorem I started asking myself **what are examples of sets of positive upper density**? It's not hard to come up with a few:

- Flip a coin with probability $\mathbb{P}(H) = p < 1$ and take all the heads
- Consider $\{ n : \{ n \theta\} > 0.001$ where $\theta$ is an irrational number

While searching the OEIS I only found a few more:

- $n$ with a prime factor such that $p > \sqrt{n}$ has density $\log 2$
- $n = p + 2^k$ where $k \geq 1$ and $p$ is prime.

Then there are square-free numbers and $\{ n : n^2 + 1 \text{ is square-free} \}$.

I have been looking through various sources... such as Roth's book who writes very abstractly. In a sense, chosen any correctly-chosen set can have positive upper density, but are there any good examples that you are aware of?

Erdos may have come up with a few. I considered making my coin-flip example deterministic.