For which infinite geometric sequences every real number can be represented as the sum of plus-minus of its terms. For what values of the quotient $q>1$ the following property holds:

Every $x\in \mathbb R$ can be represented as
$
x = \sum_{n=k}^\infty \pm q^{-n} \tag{*}\label{x}
$
for some $k\in \mathbb Z$.

The case of $q=2$ is discussed here: Every real number can be represented as a sum of plus-minus of the terms of infinite geometric sequence $2^{-n}$.
No $q>2$ has the property as then $x=0$ can not be represented in the form \eqref{x}. Indeed, suppose that \eqref{x} represents $x=0$, then by triangle inequality
\begin{align*}
0 = |x| &= \left| \sum_{n=k}^\infty \pm q^{-n} \right| \\
&\geq q^{-k} - \left| \sum_{n=k+1}^\infty \pm q^{-n} \right| \\
&\geq q^{-k} - \sum_{n=k+1}^\infty q^{-n} \\
&= q^{-k} - \frac{q^{-k-1}}{1-q^{-1}} \\
&= q^{-k} \left( 1 - \frac{1}{q-1} \right)\\
&= q^{-k} \cdot \frac{q-2}{q-1} > 0.
\end{align*}
Contradiction.
What can we tell about the case $q\in (1,2)$?
 A: We first show that if $q\in (1,2]$ then every $x\in [-1,1]$ can be written in the desired way.
Fix $q\in (1,2]$ and $x_0 \in [-1,1]$ and consider the sequence $\{x_n\}$ given recursively by
$$
  x_n = \min_{a_n \in \{-1,1\}} |x_{n-1}-a q^{-n}|,
\quad n \in \mathbb N.
$$

Lemma. $|x_n| \leq q^{-n}$.

Proof. We prove the lemma by mathematical induction:

*

*The claim holds for $n=0$ as $|x_0|\leq 1$.

*Fix $n \in \mathbb N$ and suppose that the claim holds for $n-1$, i.e. $|x_{n-1}| \leq q \cdot q^{-n}$. Choose $a_n\in \{1,-1\}$ with the same sign as $x_{n-1}$ (and $a_n=1$ when $x_{n-1}=0$), then
$$
  |x_n| = |x_{n-1}-a_n q^{-n}| = \big||x_{n-1}|- q^{-n}\big| 
\leq (q-1) q^{-n} \leq q^{-n}, 
$$
which is the claim for $n$.
$\tag*{$\Box$}$
We conclude by observing that $x_n \to 0$ as $n\to \infty$, and so
$$
  \sum_{n=1}^\infty a_n q^{-n} = x_0,
$$
which is the desired representation of $x_0$.

Finally, given any $x\in \mathbb R$, let $k\in \mathbb Z$ be any number such that $|x|\leq q^{1-k}$. Then we put $x_0=q^{k-1}x$ and consider the sequence $\{a_n\}$ representing $x_0$. We have
$$
  x = q^{k-1} x_0 = q^{k-1} \sum_{n=1}^\infty a_n q^{-n}
= \sum_{n=1}^\infty a_n q^{-n-k+1}
= \sum_{n=k}^\infty a_{n-k+1} q^{n},
$$
which is the desired representation of $x$.
