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The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces
Möbius disjointness for interval exchange transformations on three intervals
1. | Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA |
2. | Department of Mathematics, University of Chicago, Chicago, IL 60637, USA |
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.
References:
[1] |
M. Boshernitzan and A. Nogueira,
Generalized eigenfunctions of interval exchange maps, Ergodic Theory Dynam. Sys., 24 (2004), 697-705.
doi: 10.1017/S0143385704000021. |
[2] |
J. Bourgain,
On the correlation of the Moebius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.
doi: 10.1007/s11854-013-0016-z. |
[3] |
J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67â€"83.
doi: 10.1007/978-1-4614-4075-8_5. |
[4] |
H. Davenport,
On some infinite series involving arithmetical functions (Ⅱ), Quart. J. of Math., 8 (1937), 313-320.
doi: 10.1093/qmath/os-8.1.313. |
[5] |
E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue,
On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317.
doi: 10.1016/j.jfa.2013.09.005. |
[6] |
S. Ferenczi and C. Mauduit,
On Sarnak's conjecture and Veech's question for interval exchanges, J. Anal. Math., 134 (2018), 545-573.
doi: 10.1007/s11854-018-0017-z. |
[7] |
B. Green and T. Tao,
The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2012), 541-566.
doi: 10.4007/annals.2012.175.2.3. |
[8] |
I. Katai,
A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.
doi: 10.1007/BF01949145. |
[9] |
A. Khinchin, Continued Fractions, with a preface by B. V. Gnedenko, translated from the third (1961) Russian edition, reprint of the 1964 translation, Dover, 1997. |
[10] |
A. Harper, A different proof of a finite version of Vinogradov's bilinear sum inequality, online notes, 2011. |
[11] |
M. Ratner,
Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313.
doi: 10.2307/2007030. |
[12] |
D. Rudolph, Fundamentals of Measurable Dynamics. Ergodic Theory on Lebesgue Spaces, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990.
![]() ![]() |
[13] |
Z. Wang,
Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196.
doi: 10.1007/s00222-016-0707-z. |
show all references
References:
[1] |
M. Boshernitzan and A. Nogueira,
Generalized eigenfunctions of interval exchange maps, Ergodic Theory Dynam. Sys., 24 (2004), 697-705.
doi: 10.1017/S0143385704000021. |
[2] |
J. Bourgain,
On the correlation of the Moebius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.
doi: 10.1007/s11854-013-0016-z. |
[3] |
J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67â€"83.
doi: 10.1007/978-1-4614-4075-8_5. |
[4] |
H. Davenport,
On some infinite series involving arithmetical functions (Ⅱ), Quart. J. of Math., 8 (1937), 313-320.
doi: 10.1093/qmath/os-8.1.313. |
[5] |
E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue,
On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317.
doi: 10.1016/j.jfa.2013.09.005. |
[6] |
S. Ferenczi and C. Mauduit,
On Sarnak's conjecture and Veech's question for interval exchanges, J. Anal. Math., 134 (2018), 545-573.
doi: 10.1007/s11854-018-0017-z. |
[7] |
B. Green and T. Tao,
The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2012), 541-566.
doi: 10.4007/annals.2012.175.2.3. |
[8] |
I. Katai,
A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.
doi: 10.1007/BF01949145. |
[9] |
A. Khinchin, Continued Fractions, with a preface by B. V. Gnedenko, translated from the third (1961) Russian edition, reprint of the 1964 translation, Dover, 1997. |
[10] |
A. Harper, A different proof of a finite version of Vinogradov's bilinear sum inequality, online notes, 2011. |
[11] |
M. Ratner,
Horocycle flows, joinings and rigidity of products, Ann. of Math. (2), 118 (1983), 277-313.
doi: 10.2307/2007030. |
[12] |
D. Rudolph, Fundamentals of Measurable Dynamics. Ergodic Theory on Lebesgue Spaces, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990.
![]() ![]() |
[13] |
Z. Wang,
Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196.
doi: 10.1007/s00222-016-0707-z. |



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