Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat

By Jean Bourgain. 2011

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We establish small correlation bounds for the Möbius function and the Walsh system, answering affirmatively a question posed by G. Kalai [Ka]. The argument is based on generalizing the approach of Mauduit and Rivat [M-R] in order to treat Walsh functions of “large weight”, while the “small weight” case follows from recent work due to B. Green [Gr]. The conclusion is an estimate uniform over the full Walsh system. A similar result also holds for the Liouville function.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b3665b8bb398cd86ec70ea269c9c3a4f87540059

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Citations in the corpus

Measure complexity and Möbius disjointness (2017, background): “We refer to [31, 16, 5, 6, 7, 2, 29, 28, 27, 32, 3, 38, 11, 40, 21, 1, 41, 22] for the progress on this conjecture.”
Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture (2015, result): “ are uncorrelated. The conjecture is known to hold for relatively few types of dynamical systems, in particular for odometers, irrational rotations, nilsequences [GT], horocycle ows [BSZ]. See also [B1], [B], [ALR], [G], [KL], [LS], [MR], for other results. We remark, that validity of Sarnak’s conj”
Equivalence of the Logarithmically Averaged Chowla and Sarnak Conjectures (2016, unknown): “CHOWLA AND SARNAK CONJECTURES 3 dynamical systems [1], [2], [3], [5], [6], [7], [8], [9], [12], [13], [14], [15], [17], [18], [25], [29], [33], [39], [40], [41], [42], [43], [48], [49], [52], [58]; for further variants of the Sarnak conjecture, see [10], [13], [37].”
Disjointness of Moebius from Horocycle Flows (2013, background): “ element in Gand consider the discrete horocycle ﬂow (X,T), where X= Γ\Gand Tis given by T(Γx) = Γxu. (1.2) (∗)Also related to this last case is the orthogonality of µ to AC0 functions, see [K], [G], [B]. Typeset by AMS-TEX 1 Theorem 1. Let (X,T) be a horocycle ﬂow, then µis disjoint from (X,T), tha”
The logarithmic Sarnak conjecture for ergodic weights (2017, background): “The Sarnak conjecture has been proved for a variety of systems, for example nilsystems [32], some horocycle flows [6] and more general zero entropy systems arising from homogeneous dynamics [56], certain distal systems, in particular, some extensions of a rotation by a torus [45], [49], [67], a larg”
Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals (2015, unknown): “Sarnak’s conjecture has been proved to hold in several cases [1–3,6–9, 11 So far, the only automorphisms for which Sarnak’s conjecture was known to hold in each of their uniquely ergodic model were those whose prime powers are disjoint in the sense of Furstenberg [12].”