[Paper Review] On the correlation of the Moebius function with random rank-one systems

Jean Bourgain|arXiv (Cornell University)|Dec 5, 2011

Mathematical Dynamics and Fractals2 references10 citations

TL;DR

This paper establishes the Möbius disjointness property for a broad class of rank-one dynamical systems, including generalized Chacon and Katok systems, by analyzing spectral measures via generalized Riesz products and applying harmonic analysis and exponential sum estimates. It proves that the Möbius function is orthogonal to orbits of these systems, extending to a prime number theorem for such systems under mild conditions on the construction parameters.

ABSTRACT

We explore the `Moebius disjointness property' in the special context of rank-one transformations and verify this phenomenon for many of the `classical' models

Motivation & Objective

• To establish the Möbius disjointness property for rank-one dynamical systems, particularly those that are not necessarily mixing.
• To extend the orthogonality of the Möbius function to orbits of symbolic systems beyond the scope of existing MSJ (minimal self-joining) results.
• To prove a pointwise prime number theorem for systems satisfying specific structural conditions, including certain three-interval exchange transformations.
• To develop harmonic analysis techniques based on generalized Riesz products to analyze spectral measures and disjointness of powers of transformations.
• To treat both rigid and weakly mixing systems by splitting the analysis into two cases: one based on spectral singularity and another on exponential sum estimates.

Proposed method

• Uses generalized Riesz products to represent the maximal spectral type of rank-one systems, enabling spectral analysis of transformation powers.
• Applies a dichotomy: if spacers are not uniformly distributed (violating (0.7)), the system is rigid and treated via exponential sum estimates; otherwise, disjointness of T^p and T^q is shown via spectral measure decay.
• Employs the Hardy-Littlewood circle method with minor and major arcs, using restricted L1 norms of exponential sums to bound correlations with the Möbius function.
• Derives estimates for the L1 norm of exponential sums ∑ µ(n)e(nθ) against characteristic functions of words in the system’s language, achieving O(N(log N)^{-A}) decay.
• Introduces trigonometric polynomials P_j(θ) based on spacer sequences to model spectral behavior and proves that ∫ |P_j(pθ)P_j(qθ)| dθ → 0 as n → ∞.
• Combines spectral techniques with combinatorial estimates on return words in three-interval exchange systems to verify conditions for Möbius orthogonality.

Experimental results

Research questions

• RQ1 Under what conditions on the spacer sequences (an,j) and cutting parameters (wn) is the Möbius function orthogonal to orbits of a rank-one transformation?
• RQ2 Can Möbius disjointness be established for non-mixing, rigid rank-one systems such as generalized Chacon maps?
• RQ3 To what extent can exponential sum estimates and spectral analysis replace the MSJ property in proving disjointness of T^p and T^q for distinct primes p, q?
• RQ4 Can a pointwise prime number theorem be derived for symbolic systems satisfying the conditions of Theorem 2, particularly for 3-IETs?
• RQ5 What is the optimal decay rate for the L1 norm of exponential sums ∑ f(n)e(nθ) against Möbius function in systems with bounded spacers?

Key findings

• The Möbius function is orthogonal to orbits of any rank-one system satisfying wn < C and an,j < C for all n, j.
• For such systems, the L1 norm of the exponential sum ∑_{n=1}^N f(n)e(nθ) against the Möbius function decays as O(N(log N)^{-A}) for any A > 0.
• The generalized Chacon system defined by B_{n+1} = B_n^{p_n} 1 B_n^{q_n} with p_n + q_n → ∞ satisfies Möbius disjointness.
• Katok’s system, defined by B_{n+1} = B_n^{p_n} (B_n 1)^{p_n}, satisfies Möbius disjointness when p_n → ∞ sufficiently fast.
• For three-interval exchange transformations satisfying Keane’s condition and min(m_k, n_k) > C_0 for all k, the system satisfies both Möbius disjointness and a pointwise prime number theorem.
• The logarithmic factor in the error term can be removed by refining the circle method, leading to an estimate ∫ |PW| ≲ (log K)^3 Q^ε K^ε, which enables application of standard Hardy-Littlewood analysis.