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On patterns occurring in binary algebraic numbers. (English) Zbl 1151.11036

The authors prove that the binary expansion of an algebraic number contains infinitely many blocks of consecutive digits which can be expressed as \(7/3\)-powers. For instance, the block \(011010 011010 01\) is a \(7/3\)-power of the block \(011010\). Trivially, such an expansion contains infinitely many squares. Conjecturally, there are infinitely many \(\alpha\)-powers for any \(\alpha>1\).
This result seems to be the first result of this kind which goes beyond the trivial result with \(\alpha=2\). Its proof is a combination of a recent result by Adamczewski and Bugeaud on block-complexity of algebraic numbers with a result of Karhumäki and Shallit concerning infinite binary words avoiding \(7/3\)-powers. The authors also study the ternary expansion of an algebraic number and prove that it either contains infinitely many squares or infinitely many occurrences of at least one of the blocks \(010\) or \(02120\). Finally, the authors ask whether the binary expansion of every algebraic number contains arbitrarily large squares.

MSC:

11J81 Transcendence (general theory)
68R15 Combinatorics on words
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Online Encyclopedia of Integer Sequences:

Expansion of sqrt(2) in base 2.

References:

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