[Paper Review] On Multiple Recurrence

Ilya D. Shkredov|ArXiv.org|Jun 21, 2004

Fixed Point Theorems Analysis1 references3 citations

TL;DR

This paper establishes a quantitative version of Furstenberg's multiple recurrence theorem for dynamical systems with two commuting measure-preserving transformations. By introducing a novel recurrence function involving Hausdorff-type measures and entropy, it proves that for almost every point, the joint recurrence time is bounded by a function of the metric entropy, yielding explicit upper bounds on the density of sets without corners in Z² and improving on previous logarithmic-type bounds for Szemerédi-type problems.

ABSTRACT

Let X be a metric space with metric d and T,S be two commutative measure-preserving maps of X. In this paper we obtain numerical results about multiple recurrence of almost every point of this dynamical system. On other words we study the question about convergence to zero of max{d(T^n x,x), d(S^n x,x)}.

Motivation & Objective

• To extend Furstenberg's multiple recurrence theorem to systems with two commuting measure-preserving transformations.
• To provide quantitative bounds on the recurrence time for such systems, improving on qualitative ergodic results.
• To apply these bounds to estimate the maximal density of corner-free subsets in the two-dimensional grid [1,N]².
• To derive explicit upper bounds on the function L(N), measuring the maximal density of sets without corners, using recurrence theory.

Proposed method

• Introduces a recurrence function C_{S,R}(x) involving the liminf of n·h(d(Sⁿx,x)) and n·h(d(Rⁿx,x)) for a continuous, increasing function h.
• Uses Hausdorff-type measures H_h and ε-entropy N_ε(X) to quantify the size of recurrence sets.
• Applies a covering lemma (Lemma 2.10) to bound the measure of the set Y(t) where both S and R avoid returning to Y within t steps.
• Establishes a Stieltjes integral inequality (Theorem 2.8) to relate the distribution of recurrence times to the measure of sets.
• Uses the existence of invariant measures on compact metric spaces (Lemma 2.11) to ensure the applicability of the recurrence framework.
• Applies the main recurrence bound to the corner problem in Z², deriving a quantitative upper bound on L(N).

Experimental results

Research questions

• RQ1 What is the quantitative rate at which multiple recurrence occurs in systems with two commuting transformations?
• RQ2 How can the recurrence time for joint orbits be bounded using metric and measure-theoretic tools?
• RQ3 What is the maximal possible density of a subset of [1,N]² with no corner triple, and how fast does this density decay as N increases?
• RQ4 Can the ergodic recurrence result of Furstenberg be strengthened to yield explicit bounds in terms of entropy or Hausdorff measure?
• RQ5 How does the joint recurrence behavior of two commuting maps relate to the existence of arithmetic configurations in sparse sets?

Key findings

• The paper proves that for any set A ⊆ [1,N]² of density at least δ, if N ≥ exp(exp(exp(δ⁻ᶜ))) for an absolute constant c > 0, then A must contain a corner triple.
• It establishes that L(N) ≤ 100 / log_*¹ᐟ⁴ N, improving upon earlier bounds for the corner problem.
• The main recurrence result yields L(N) ≪ 1 / (log log log N)^{C₁} for an absolute constant C₁ > 0, providing a quantitative version of Furstenberg's theorem.
• For a compact metric space X with H_h(X) < ∞ and two commuting maps S, R, there exists a point x such that liminf_{n→∞} L⁻¹(n) · max{h(d(Sⁿx,x)), h(d(Rⁿx,x))} ≤ C for some absolute constant C.
• The function C_{S,R}(x) is μ-integrable and satisfies ∫_A C_{S,R}(x) dμ ≤ H_h(A), providing a measure-theoretic control on recurrence.
• The bound μ(Y(t)) ≤ L(t) for the set of points avoiding recurrence in both maps up to time t is derived via a covering argument on the grid [1,t]².