[Paper Review] Möbius Disjointness for Nilsequences Along Short Intervals

Xiaoguang He, Zhiren Wang|arXiv (Cornell University)|May 8, 2019

Analytic Number Theory Research13 references4 citations

TL;DR

This paper establishes a quantitative bound on the Möbius disjointness of nilsequences along short intervals, proving that the correlation between the Möbius function and a 1-Lipschitz nilsequence decays uniformly as both the interval length $ H $ and the total range $ N $ tend to infinity. The result generalizes Davenport's estimate and Green-Tao's orthogonality to short intervals, using effective equidistribution and minor arc analysis for polynomial sequences in nilmanifolds.

ABSTRACT

For a nilmanifold $G/Γ$, a $1$-Lipschitz continuous function $F$ and the Möbius sequence $μ(n)$, we prove a bound on the decay of the averaged short interval correlation $$\frac1{HN}\sum_{n\leq N}\Big|\sum_{h\leq H} μ(n+h)F(g^{n+h}x)\Big|$$ as $H,N o\infty$. The bound is uniform in $g\in G$, $x\in G/Γ$ and $F$.

Motivation & Objective

• To establish a quantitative, uniform bound on the short interval correlation between the Möbius function and nilsequences in nilmanifolds.
• To extend Davenport’s classical estimate on exponential sums to nilsequences and short intervals.
• To unify the methods of Green-Tao (orthogonality of Möbius to nilsequences) and Matomäki-Radziwiłł (short interval averages of multiplicative functions).
• To provide an effective, uniform decay rate independent of the nilmanifold structure, group element, or observable function.

Proposed method

• Uses a quantitative factorization theorem for 2-parameter polynomials to decompose the nilsequence into structured and pseudorandom parts.
• Applies a separation of major and minor arcs, with major arcs handled via equidistribution estimates and minor arcs via the Kátai-Bourgain-Sarnak-Ziegler criterion.
• Employs a modified version of the Matomäki-Radziwiłł short interval method, incorporating effective equidistribution for polynomial orbits in nilmanifolds.
• Introduces a hybrid major-minor arc estimate using the function $ ilde{M}(eta, X, Y) $, which quantifies the cancellation in multiplicative functions.
• Applies Vaughan’s identity and bilinear methods to control the minor arc contribution, leveraging bounds from [MRT16].
• Constructs an exceptional set $ ilde{\mathcal{S}} $ of density $ \ll \epsilon N $, outside of which the correlation is small, and controls the error via $ \epsilon $-dependent bounds.

Experimental results

Research questions

• RQ1 Can the Möbius disjointness principle be extended from long to short intervals for nilsequences?
• RQ2 What effective, uniform decay rate can be achieved for the short interval correlation $ \left| \sum_{h \leq H} \mu(n+h) F(g^{n+h}x) \right| $ as $ H, N \to \infty $?
• RQ3 How can the methods of Green-Tao (long interval orthogonality) and Matomäki-Radziwiłł (short interval averages) be combined for nilsequences?
• RQ4 Is the decay uniform over all nilmanifolds, group elements, and 1-Lipschitz observables?

Key findings

• The paper establishes a uniform bound of the form $ \frac{1}{HN} \sum_{n \leq N} \left| \sum_{h \leq H} \mu(n+h) F(g^{n+h}x) \right| \ll \left( H^{-\epsilon} + H^{4C\epsilon} e^{-\frac{1}{2} \tilde{M}(\beta, N/H^{10C\epsilon}, H^{2C\epsilon})} \tilde{M}(\cdots)^{1/2} + H^{4C\epsilon} (\log(N/H^{10C\epsilon}))^{-1/100} \right) $, with $ \epsilon > 0 $ arbitrary and $ C $ depending only on the nilpotency class and dimension.
• The bound decays to zero as $ N \to \infty $, uniformly in $ g \in G $, $ x \in G/\Gamma $, and 1-Lipschitz $ F $, under the condition $ N > \exp((\log H)^2) $.
• The decay is effective and quantitatively controlled via the function $ \tilde{M}(\beta, X, Y) $, which tends to infinity as $ X \to \infty $ for the Möbius function $ \beta = \mu $, ensuring the error terms vanish.
• The result implies that the average short interval correlation tends to zero, confirming a strengthening of the Möbius disjointness conjecture for nilsystems in short intervals.
• The exceptional set $ \mathcal{S} $, where the bound may fail, has density $ \ll \epsilon N $, and the error from this set is controlled by $ \epsilon $, which can be made arbitrarily small.