# Split Question: Bergelson Multiplicative Density of “Even-like” sets

This post splits the post: https://math.stackexchange.com/questions/1026755/questions-about-bergelson-multiplicative-upper-density?noredirect=1#comment2093891_1026755 into one more concentrated series of questions. It is largely copied directly.

Let $\mathbb{P} \subset \mathbb{N}$ be the set of all primes. Let $\forall n \in \mathbb{N}, F_n = \{a_n \prod_{i=1}^n (p_i^{r_i}): a_n \in \mathbb{N}, p_i \in \mathbb{P}, r_i \in [0, N_i(n)] \cap \mathbb{Z} \forall i \in \mathbb{N} \cap [1, n] \}$ where $\forall i \in \mathbb{N} \cap [1, n], N_i(n) \to \infty$. (For concreteness, you can take $N_i(n) = n \forall i$, but please mention when you do and if it actually matters. Likewise, $a_n = 1 \forall n$ is allowed, but just mention it when it comes up.) Denote the Bergelson multiplicative upper density of the set $A$ by $\overline{\operatorname{d}^{\times}}(A) = \limsup_{N \to \infty} \big(\frac{A \cap F_n}{|F_n|} \big)$. Let $\operatorname{d}$ denote the additive natural density.

Many of the following are my speculations. But, being a novice to this notion of density (I do not yet even have a great grasp of what it intuitively means, although I am working on it), I am having difficulties proving or disproving these claims. I am mostly just feeling things out at the moment.

Question 1: $\overline{\operatorname{d}^{\times}}(2 \mathbb{N} - 1) =^? 0$?

Question 2: Let $E = \{\prod_{i=1}^k (p_i^{r_i}) < \infty: p_i \in \mathbb{P} \forall i \in \mathbb{N}, \sum_{i=1}^k (r_i) \in 2 \mathbb{N} \}$. Then $\overline{\operatorname{d}^{\times}}(E) =^? \frac{1}{2}$?

Question 3: Let $\tilde{E}_k = \{\prod_{i=1}^k (p_i^{r_i}) < \infty: \forall i \in \mathbb{N}, p_i \in \mathbb{P}, r_i \in 2 \mathbb{N} \}$. What is $\overline{\operatorname{d}^{\times}}(\tilde{E}_k)$?
Bonus 1: Prove that $\operatorname{d}(E) = \frac{1}{2}$.
1.) I know that the set of odd numbers is, obviously, missing 2 and any number that it divides while it also contains every other prime power. Thus, there are a lot of elements of $F_n$ that are missing in $2 \mathbb{N} - 1$, but it, at this time, is not readily apparent that most of them are in fact missing. Would the same be true if we took instead the set of all prime powers that are not divisible by $3$ (or any specific given prime)? I do not see why not. But, if that is so, then how [in some way] is $\lim_{n \to \infty} \big( \frac{|2 \mathbb{N} \cap F_n|}{|F_n|} \big) =^? 1$ special (presuming that it is even correct; see Question 4 in the initially linked post)? Are multiples of powers of 2 are almost all of the numbers in this sense? That does not seem very obvious to me. But I do guess that half of all natural numbers are even, while less are divisible by other given primes- I still do not see why $2 \mathbb{N}$ would be basically everything though. [I originally asked Question 4 along with Question 1 in our to pointedly demonstrate my hiccup in understanding in this area. I do not know whether or not the resolution to Question 4 is affirmative, but if it is, I understand less than I would like. An explanation here, in particular, would be much appreciated.]