So, I have read this problem, and it's bugged me since:
Let $a \in (1,2)$ be a transcendental number
1) Let $Y = \{ P(a) : P \in \mathbb{Z}[X] \}$, show that Y is dense in $\mathbb{R}$.
2) Let $X = \{ P(a) : P \in \mathbb{Z}[X], P \text{ has coefficients in } \{0,1,-1\}\}$, show that X is dense in $\mathbb{R}$.
I have been able to prove 1), by showing that $Y$ is a non-cyclic subgroup of $\mathbb{R}$ and therefore is dense (a well-known result which I can prove too).
2) had a hint attached:
Hint: Start by proving that $0$ is in the closure of $X \backslash \{0\}$
I was able to prove this too:
- I took $Z = \{ \sum_{k=0}^{n-1} b_k a^k / \forall k, b_k \in \{0, 1\} \}$.
- We have $\forall z \in Z, 0 \leq z \leq \frac{a^n - 1}{a-1} < \frac{a^n}{a-1}$.
- $Z$ has $2^n$ elements, so if we cut $[ 0, \frac{a^n}{a-1} )$ in the $2^n - 1$ intervals $I_k = [ \frac{(a/2)^n}{a-1} \times k, \frac{(a/2)^n}{a-1} \times (k+1) )$, then there is a $k$ such that $I_k \cap Z = \{z_1, z_2\}$ with $z_1 \neq z_2$.
- Therefore, $| z_1 - z_2 | \in X \cap [0, \frac{(a/2)^n}{a-1})$. Since $0 < a < 2$, $(\frac{a}{2})^n \rightarrow 0$ and we have that X is dense around 0.
But now I'm stuck. I've thought of trying to approximate any real by an element of the form $a^n \varepsilon$ with $\varepsilon \in X$ close to 0, but it doesn't work. Do you have any hints?
Edit: It's not homework, it's in preparation of an upcoming oral exam.