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Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational rotation map) tells us a lot about the initial digits. But when $n$ is large, most digits of $2^n$ are neither initial digits nor final digits. Do we currently have any tools that would let us draw nontrivial conclusions about the statistical properties of the sequence obtained by concatenating the decimal expansions of the successive powers of two (https://oeis.org/A000455)? It’s easy to show that every digit string occurs infinitely often, but do we know (for instance) that asymptotically each decimal digit occurs 1/10 of the time?

In a similar spirit: There is no reason to think that, from some point onward, sequence A000455 agrees with sequence A000796 (the sequence of successive digits of pi), but I do not see a way to disprove this unlikely hypothesis. Can anyone show that the sequences disagree infinitely often?

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    $\begingroup$ There are many open problems in this area, e.g., erdosproblems.com/406 or en.wikipedia.org/wiki/Mahler%27s_3/2_problem . Adjacent powers of two have some relations between digits, of course, which can be used to derive some consequences, but in general these questions are quite difficult. As a general rule, we know quite little about the distribution of fractional parts of specific exponentially growing sequences. $\endgroup$
    – Terry Tao
    Dec 20 at 2:43
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    $\begingroup$ it's believed, but not proved, that $2^{86}$ is the biggest power of two with no zeros in its decimal representation. oeis.org/A007377 $\endgroup$
    – Gerry Myerson
    Dec 20 at 17:25

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