Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets

2026/07/17 by Félix Brokering Pinilla, Alex Iosevich, Ben Krause
#math.DS #math.CA #math.PR

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Abstract

Let d ≥ 3, D \subsetneq \ 0,1,…,d-1\, |D| ≥ 2, 0 ∈ D be a finite alphabet, and define the integer Cantor set C := C_D := \bigcup_{J ≥ 0} \ ∑_{j =0}^J a_j d^j : a_j ∈ D \. We prove that for any σ-finite measure-preserving system, (X,μ,T), and any f ∈ L^p(X), 2≤ p<∞, the ergodic averages (1)/(|C_N|) ∑_{n ∈ CN} f(Tⁿ x), C_N := C ∩ \1,2,…,N \ converge μ-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if C' := \ ∑_{j ≥ 1} a_j d^-j : a_j ∈ D \ ⊂ [0,1] is a real-variable Cantor set, and ν denotes its natural measure, then we prove that lim_{k → ∞} ∫ f(x-d^-k t) dν(t) = f(x) Lebesgue almost-everywhere for any f ∈ L²_loc(ℝ).

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