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Möbius disjointness for interval exchange transformations on three intervals

JC: Supported in part by NSF grants DMS-135500 and DMS-1452762 and the Sloan foundation.
AE: Supported in part by NSF grant DMS 1201422 and the Simons Foundation.
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  • We show that Sarnak's conjecture on Möbius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

    Mathematics Subject Classification: Primary: 37E05; Secondary: 28D20.


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  • Figure 1.  The torus $\hat{X}$. A vertical segment of length $q_k$ intersects a horizontal slit of length $z$

    Figure 2.  The torus $g_{\log(q_k)} \hat{X}$: A vertical segment $\gamma_1$ of length $1$ (drawn in red) intersects a horizontal slit $\gamma_2$ of length $q^k z$ (drawn in blue)

    Figure 3.  Closing the curves. We complete the vertical segment $\gamma_1$ to a closed curve $\hat{\gamma_1}$ by adding a horizontal segment $\zeta_1$ (drawn in green). Simularly, we close up the horizontal slit $\gamma_2$ to obtain a closed curve $\hat{\gamma}_2$ by adding in a horizontal segment $\zeta_2$ and a vertical segment $\zeta_2'$ (drawn in purple)

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