Short version of my question: I'm interested in the following fact.

If $m,n$ are odd integers, then $m/n$ can be written as the ratio of two numbers of the form $\sum_{j=0}^\ell \epsilon_j 4^j$, where $\epsilon_j \in \{-1,0,1\}$.

More background:

Let $A = \{0,1,4,5,16,\ldots\}$ be the set of nonnegative integers whose base 4 expansion contains only the digits 0 and 1.

Given a number $r \in \mathbb Q \setminus \{0\}$, there is a unique way (up to sign) to write $r = 4^k \frac{m}{n}$, where (1) $k \in \mathbb Z$, (2) $m$ and $n$ are not divisible by 4, and (3) $\gcd(m, n) = 1$.

Then the following is true:

Theorem:Let $r \in \mathbb Q \setminus \{0\}$. Then $r \in \frac{A-A}{A-A} = \{ \frac{a-b}{c-d} ~|~ a,b,c,d \in A\}$ if and only if both $m$ and $n$ are odd.

The "only if" direction can be proved easily with elementary number theory. (Every nonzero element of $A-A$ contains an even number of factors of $2$.)

For the "if" direction, the only proof I know is in Section 10.3 of Mattila's *Fourier Analysis and Hausdorff Dimension* (and is based on Kenyon's 1997 paper *Projecting the one-dimensional Sierpinski gasket*). The proof looks at the Fourier transforms of measures defined on projections of the four-corner Cantor set. Actually, the textbook proves something else, and the theorem I stated above is a corollary. This leads me to wonder if there is an easier proof of the "if" direction of the theorem above (e.g., via elementary number theory).