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Published online by Cambridge University Press: 24 October 2008
Let α be an irrational p-adic number, r an arbitrary positive integer. Our aim is to prove that there exists a rational integer X satisfying
such that every possible sequence of r digits 0, 1, …, p – 1 occurs infinitely often in the canonical p-adic series for Xα. It is clear that it suffices to prove this result for p-adic integers.