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Abstract

We establish independence properties of continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational real algebraic numbers have the same long subword, then the two expansions have the same tails. If the two expansions have mirror symmetric long subwords, then both the algebraic numbers are quadratic. Applying the above results, we prove a theorem analogous to Roth’s theorem about approximation by algebraic numbers.