Fractional parts of powers and Sturmian words
Yann Bugeaud, Artūras Dubickas · Comptes Rendus Mathématique · 2005
Let b ⩾ 2 be an integer. In terms of combinatorics on words we describe all irrational numbers ξ > 0 with the property that the fractional parts { ξ b n } , n ⩾ 0 , all belong to a semi-open or an open interval of length 1 / b . The length of such an interval cannot be smaller, that is, for irrational ξ , the fractional parts { ξ b n } , n ⩾ 0 , cannot all belong to an interval of length smaller than 1 / b .