Uniform Weighted Averages and a Conjecture of Bergelson, Moreira, and Richter

2026/02/24 by Michael Reilly
#math.DS

paper · pdf · doi:10.1017/etds.2026.10325

Abstract

We confirm a conjecture posed by Bergelson, Moreira, and Richter (arXiv:1711.05729), and in particular show that for every probability measure preserving system (X,\mathscrB,μ,T), every k∈ ℕ, every set A∈ \mathscrB with μ(A)>0, and every tempered function f, lim_N→∞(1)/(N)∑_n=1^Nμ(A∩ T^-\lfloorf(n)\rfloorA∩ T^-\lfloorf(n+1)\rfloorA∩ ⋯ ∩ T^-\lfloorf(n+k)\rfloorA)>0. This is achieved by establishing conditions on an increasing function W:ℕ→ (0,∞) such that if (x_n)_{n∈ ℕ} is a bounded sequence in a Banach space with lim_{W(N)-W(M)→∞}(1)/(W(N)-W(M))∑_n=M^N (W(n)-W(n-1))x_n =L then the limit of Cesàro averages of (x_n)_{n∈ ℕ}, lim_N→∞(1)/(N)∑_n=1^Nx_n is also equal to L. Furthermore, the methods we develop can be used to sharpen some of the combinatorial results obtained by Bergelson, Moreira, and Richter. For example, if E is a set of positive upper density, then for any k∈ ℕ, any ε>0, and all sufficiently large N∈ ℕ there is an n∈ [N-N^1/2+ε,N] such that \a,a+\lfloorn^3/2\rfloor,a+\lfloor(n+1)^3/2\rfloor,… ,a +\lfloor(n+k)^3/2\rfloor\⊆ E.

Citations

• Furstenberg systems of Hardy field sequences and applications
• Uniform distribution and Hardy fields

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