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We give an optimal version of the classical ``three-gap theorem'' on thefractional parts of $n \theta$, in the case where $\theta$ is an irrationalnumber that is badly approximable. As a consequence, we deduce a version ofKronecker's inhomogeneous approximation theorem in one dimension for badlyapproximable numbers. We apply these results to obtain an improved measure ofsequence diversity for characteristic Sturmian sequences, where the slope isbadly approximable.

Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences