Dexter
Here are my thoughts let me know what you think.
The ternary expansion of LRLRLRLRLR...... is .0202020202......
Which we can see would look something like $$0\cdot1/3 +2\cdot1/9+0\cdot1/27+2\cdot1/81.....$$
Which we can write as $$\sum\limits_{n=1}^\infty 2\cdot(1/3)^{2n} =2\cdot\sum\limits_{n=1}^\infty ((1/3)^{2})^n=2\cdot\sum\limits_{n=1}^\infty (1/9)^{n}=2\cdot\sum\limits_{n=0}^\infty (1/9)^{n+1}$$
Then after we shift the index $$2\cdot\sum\limits_{n=0}^\infty (1/9)^{n}\cdot(1/9)=(2/9)\cdot\sum\limits_{n=0}^\infty (1/9)^{n}$$
Then using the geometric series expansion we get $$\frac{2}{9}\cdot\frac{1}{1-\frac{1}{9}}=\frac{2}{9}\cdot\frac{9}{8}=\frac{1}{4}$$
Thus we see $\frac{1}{4}$ is in the Cantor set.
