Proof that a block of digits doesn't repeat twice in a row in an irrational number in binary

Question

So I've been trying to figure out this problem for 3 hours now and don't really know how to start.

What I'm trying to do is figure out if there is guaranteed to be, in an irrational number written in binary, a block of binary digits of length greater than once that repeats twice in a row so for example: $0.\underline{110}-\underline{110}$ has the block 110 twice. I know it's most probably true since I verified it using programming, unless something is wrong with my code, but I want to prove it without using the computer, to get some more insight about it.

What I've tried doing is write out all possible combinations for 4 digits and try out different combinations etc, I thought I'd have some idea well before I'm quarter way through combinations of a pair of 4 digit block, but that turned out not to be the case and there was a lot of wasted time on just trial and error.

I just don't know how to start on this one, any hints/ proofs would be very appreciated.

If needed I have studied a first course in number theory so I can understand that.

Answer 1

In the paper "On nonrepetitive sequences" by Entringer and Jackson, in J. Combin. Theory Ser. A 11 (1974), 159–164. The link is http://www.sciencedirect.com/science/article/pii/0097316574900417, the authors show that any binary sequence of length more than 18 must have two identical consecutive blocks, the length of which is at least 2.

The proof is by case analysis, shown in Figure 1 in the paper.

Answer 2

You can deliberately define an irrational number to have this non-repeating quality, if required. For example, for an irrational number that avoids 3-blocks of repeating digits, take the binary definition of $\pi$ and generate a new number such that each digit $x$ in $\pi$ is replaced by $11x00$. (This also avoids 4-blocks).

To avoid any repetitions of whatever length, you would probably have to be more extravagant, coding further digits into ever larger strings (possibly using transforms of earlier digits). My expectation is that it is possible though.