Möbius disjointness for nilsequences along short intervals

He, Xiaoguang; Wang, Zhiren

doi:10.1090/tran/8176

Möbius disjointness for nilsequences along short intervals
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by Xiaoguang He and Zhiren Wang PDF

Trans. Amer. Math. Soc. 374 (2021), 3881-3917 Request permission

Abstract:

For a nilmanifold $G/\Gamma$, a $1$-Lipschitz continuous function $F$ and the Möbius sequence $\mu (n)$, we prove a bound on the decay of the averaged short interval correlation \begin{equation*} \frac {1}{HN}\sum _{n\leq N}\Big |\sum _{h\leq H} \mu (n+h)F(g^{n+h}x)\Big | \end{equation*} as $H,N\to \infty$. The bound is uniform in $g\in G$, $x\in G/\Gamma$ and $F$.

References

J. Bourgain, P. Sarnak, and T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 67–83. MR 2986954, DOI 10.1007/978-1-4614-4075-8_{5}
H. Davenport, On some infinite series involving arithmetical functions II, Quat. J. Math. 8 (1937), 313–320.
El Houcein el Abdalaoui, Mariusz Lemańczyk, and Thierry de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, Int. Math. Res. Not. IMRN 14 (2017), 4350–4368. MR 3674173, DOI 10.1093/imrn/rnw146
Sébastien Ferenczi, Joanna Kułaga-Przymus, and Mariusz Lemańczyk, Sarnak’s conjecture: what’s new, Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Lecture Notes in Math., vol. 2213, Springer, Cham, 2018, pp. 163–235. MR 3821717
Livio Flaminio, Krzysztof Frączek, Joanna Kułaga-Przymus, and Mariusz Lemańczyk, Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds, Studia Math. 244 (2019), no. 1, 43–97. MR 3850672, DOI 10.4064/sm170512-25-9
Nikos Frantzikinakis and Bernard Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math. (2) 187 (2018), no. 3, 869–931. MR 3779960, DOI 10.4007/annals.2018.187.3.6
Andrew Granville and K. Soundararajan, Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 357–384. MR 2276774, DOI 10.1090/S0894-0347-06-00536-4
Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), no. 2, 465–540. MR 2877065, DOI 10.4007/annals.2012.175.2.2
Ben Green and Terence Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), no. 2, 541–566. MR 2877066, DOI 10.4007/annals.2012.175.2.3
G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403 (German). MR 230694, DOI 10.1007/BF01894515
Bingrong Huang, Strong orthogonality between the Möbius function and nonlinear exponential functions in short intervals, Int. Math. Res. Not. IMRN 23 (2015), 12713–12736. MR 3431634, DOI 10.1093/imrn/rnv091
Bingrong Huang, Exponential sums over primes in short intervals and an application to the Waring-Goldbach problem, Mathematika 62 (2016), no. 2, 508–523. MR 3521339, DOI 10.1112/S0025579315000352
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar. 47 (1986), no. 1-2, 223–225. MR 836415, DOI 10.1007/BF01949145
Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190 (French). MR 0088496
A. Leibman, Polynomial sequences in groups, J. Algebra 201 (1998), no. 1, 189–206. MR 1608723, DOI 10.1006/jabr.1997.7269
A. Leibman, Polynomial mappings of groups, Israel J. Math. 129 (2002), 29–60. MR 1910931, DOI 10.1007/BF02773152
A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 201–213. MR 2122919, DOI 10.1017/S0143385704000215
A. Leibman, Orbit of the diagonal in the power of a nilmanifold, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1619–1658. MR 2563743, DOI 10.1090/S0002-9947-09-04961-7
A. I. Mal′cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9–32 (Russian). MR 0028842
Kaisa Matomäki and Maksym Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), no. 3, 1015–1056. MR 3488742, DOI 10.4007/annals.2016.183.3.6
Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao, An averaged form of Chowla’s conjecture, Algebra Number Theory 9 (2015), no. 9, 2167–2196. MR 3435814, DOI 10.2140/ant.2015.9.2167
Kaisa Matomäki and Xuancheng Shao, Discorrelation between primes in short intervals and polynomial phases, to appear in Int. Math. Res. Not. (2019).
Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, DOI 10.1215/S0012-7094-91-06311-8
Peter Sarnak, Three lectures on the Möbius function, randomness and dynamics, lecture notes, IAS (2009).
Nimish A. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Lie groups and ergodic theory (Mumbai, 1996) Tata Inst. Fund. Res. Stud. Math., vol. 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 229–271. MR 1699367
Terence Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi 4 (2016), e8, 36. MR 3569059, DOI 10.1017/fmp.2016.6
Terence Tao and Joni Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, Duke Math. J. 168 (2019), no. 11, 1977–2027.
R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge Tracts in Mathematics, vol. 125, Cambridge University Press, Cambridge, 1997. MR 1435742, DOI 10.1017/CBO9780511470929
Tao Zhan, On the representation of large odd integer as a sum of three almost equal primes, Acta Math. Sinica (N.S.) 7 (1991), no. 3, 259–272. A Chinese summary appears in Acta Math. Sinica 35 (1992), no. 4, 575. MR 1141240, DOI 10.1007/BF02583003