Golden ratio base

Question

Let $\phi$ be the golden ratio and look at real numbers as expansions in digits from base $\phi + 1$. Has this base been considered or studied anywhere?

Note that integers in this base are palindromes with the dot/comma after the middle digit. My idea was to use this base for data storage since it's closer to the most economical choice of radix ($e$) than $3$ for storage of arbitrary reals but integers also have a short representation. The rule for when a digit is too large is also simple (a sum of a sequence of $n$ digits may not exceed $n+1$).

So we count $1$, $2$, $10.1$, $11.1$, $12.1$, $20.2$, $100.01$ etc.

Answer 1

Not an answer but a long comment:

Such integer representations give rise to a nice language $\mathcal L$ in $\mathbb Z^*$ (finite words with letters in $\mathbb Z$). A finite word $w=x_1\ldots x_l$ belongs to $\mathcal L$ if and only if $w$ has no pair of successive strictly negative letters and the last letter $x_l$ satisfies $x_l\geq -1$. Requiring moreover $x_1\not=0$ (corresponding to no leading $0$'s) yields a bijection between $\mathcal L$ and $\mathbb N$.

The $(1+\phi)$-representation associated to such a word $x_1\ldots x_l$ is obtained by replacing $x_i\geq 0$ by $01^{x_i}$ (a $0$ followed by $x_i$ letters $1$) and by replacing $x_i<0$ by $21^{-x_i-1}$ (a letter $2$ followed by $\vert x_i\vert-1$ letters $1$).

The resulting word in $\lbrace 0,1,2\rbrace^*$ is the truncation at the point of the symmetric representation in base $\phi+1$ of a suitable integer.

Example: Substitution in the word $(3)(0)(2)(-2)(0)(0)(-3)(0)(-1)$ of $\mathcal L$ leads to $1110011210021102$ corresponding to the integer $2868871$ with $(1+\phi)$-expansion given by $1110011210021102.011200121100111$.

Perhaps someone has studied some language similar to $\mathcal L$?