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Find an irrational number whose binary development does not contain strings of $0$s or $1$s of arbitrary length.

I found it hard to find such an example since the only two digits we can use are $0$ and $1$. How can we construct such an example?

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    What do you mean by "does not contain strings of arbitrary length"? It means that its binary expansion has only finitely many digits (but in such a case it would be rational!)? – 2017-01-07
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    @Crostul I mean like strings of $1$s: $11111\ldots$ or $0$s: $00000\ldots$. – 2017-01-07

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Consider $0.\overline{10}_2$ and change every $0$ in position $(2n)!$ after the dot with a $1$.

Such binary representation is aperiodic (so it gives an irrational number) and given by a concatenation of $0$, $1$ and $111$ only. Additionally, such number is trascendental, as the sum between $\frac{2}{3}$ and a Liouville number.

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Call $a=101$, $b=1001$. Using only these two strings as "digits" write a non-repeating infinite pattern, for example $$0.abaabaaabaaaab...$$ and then substitute $a=101$, $b=1001$.

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The following binary expansion contains no substring $11$ or longer and no substring $000$ or longer:

$$0.01\color{crimson}00101\color{crimson}0010101\color{crimson}001010101\color{crimson}0\ldots$$

It’s not eventually periodic, so it’s irrational.

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    Why does this represent an irrational number? – 2017-01-07
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    @Servaes: Because it's not eventually periodic. (I didn’t bother to mention that originally, because I thought it evident.) – 2017-01-07
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Take an irrational number (e.g. $\sqrt 2$) in binary format, and replace each $1$ with the sequence $011$ and each $0$ with the sequence $001$. Then you don't have more than two consecutive $0$'s or $1$'s and the generated number is also irrational (does not have periodic substrings).

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The number $$.1?0?1?0?1?0?\dots$$ contains no string of $0$s or $1$s of length $4$. There are uncountably many ways to give values to the $?$ marks, most of them resulting in irrational numbers.

Better, the Thue–Morse sequence represents an irrational number and has no strings of length $3$: $$.0110100110010110\dots$$

Or the Kolakoski sequence (with $2$s changed to $0$s): $$.10011010010011011\dots$$