For every growth majorant strictly below exp(c/(1-|z|)), there exists a positive-measure interval-free set with the Khinchin-Ostrowski property; at or above that threshold, containing an interval becomes necessary.
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A critical majorant for the Khinchin-Ostrowski property
For every growth majorant strictly below exp(c/(1-|z|)), there exists a positive-measure interval-free set with the Khinchin-Ostrowski property; at or above that threshold, containing an interval becomes necessary.