Measure of a subset of $[0,1]$

Question

$A\subset[0,1]$ is set of all numbers with binary representation $0.c_1c_2c_3...$ where $c_i=0,1$ for all $i$ and $c_{i-1}c_{i+1}=0$ for all even $i$. How do I show that $A$ is zero measure set?

Answer 1

Let $F$ be the set of binary $4$-tuples $(a,b,c,d)$ such that one of the following holds: i) $a =0,$ or ii) $a = 1$ and $c = 0.$ Note that there are exactly $12$ distinct $4$-tuples in $F.$

Define $A'$ to be the set of $a\in [0,1]$ whose binary expansion $.a_1 a_2 \cdots$ is such that all of the $4$-tuples $(a_1,a_2,a_3,a_4),$ $(a_5,a_6,a_7,a_8),$ $(a_9,a_{10},a_{11},a_{12}),\dots$ belong to $F.$

Then $A\subset A'.$ Thus it's enough to show $m(A') = 0.$

Now $m(2^4A') = 2^4m(A').$ But note that because of the periodic nature of $A',$ $2^4A'$ is the pairwise disjoint union of $12$ sets, each of which has the form $n+A',$ where $n$ is a positive integer. Thus

$$2^4m(A') = 16m(A') = 12m(A').$$

It follows that $m(A')=0$ as desired.