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A canonical thickening of ℚ and the entropy of α-continued fraction transformations

Published online by Cambridge University Press:  14 September 2011

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (email:
Department of Mathematics, Harvard University, 1 Oxford Street, 02138 Cambridge, MA, USA (email:


We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued fractions, thus proving a conjecture of Nakada and Natsui.

Research Article
Copyright © Cambridge University Press 2011

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