Disjunctivity of Power Concatenations in Arbitrary Bases

This paper studies the concatenation of the base-b representations of the powers a, a², a³, and so on, where a and b are integers greater than or equal to 2. It proves a complete classification: the concatenation is disjunctive if and only if log base b of a is irrational, equivalently, if and only if a and b are multiplicatively independent. When the logarithm is irrational, Weyl equidistribution shows that every prescribed digit block with a nonzero first digit occurs as the beginning of infinitely many powers. This implies that every finite digit word occurs infinitely often in the concatenation. When the logarithm is rational, a common-power relation between a and b forces all sufficiently large powers to consist of one nonzero base-b digit followed by zeros, which prevents disjunctivity. The paper relates this classification to classical results on uniform distribution and leading digits. It distinguishes disjunctivity from the stronger property of normality and does not claim that the underlying Weyl or Benford principles are new.