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Preprints, Working Papers, ... Year : 2023

Generation of measures on the torus with good sequences of integers

Distribution d'une rotation le long d'une sous-suite


Let S := (s_1 < s_2 < . . . ) be a strictly increasing sequence of positive integers and denote e(β) := e^{2πiβ}. We say S is good if for every real α the limit lim_N 1/N ∑n≤N e(s_nα) exists. By the Riesz representation theorem, a sequence S is good iff for every real α the sequence (s_nα) possesses an asymptotic distribution modulo 1. Another characterization of a good sequence follows from the spectral theorem: the sequence S is good iff in any probability measure preserving system (X, m, T) the limit lim_N 1/N ∑n≤N f (T^{s_n} x) exists in L^2-norm for f ∈ L^2(X). Of these three characterization of a good set, the one about limit measures is the most suitable for us, and we are interested in finding out what the limit measure μ_{S,α} := lim_N 1/N ∑n≤N δ_{s_nα} on the torus can be. In this first paper on the subject, we investigate the case of a single irrational α. We show that if S is a good set then for every irrational α the limit measure μ_{S,α} must be a continuous Borel probability measure. Using random methods, we show that the limit measure μ_{S,α} can be any measure which is absolutely continuous with respect to the Haar-Lebesgue probability measure on the torus. On the other hand, if ν is the uniform probability measure supported on the Cantor set, there are some irrational α so that for no good sequence S can we have the limit measure μ_{S,α} equal ν. We leave open the question whether for any continuous Borel probability measure ν on the torus there is an irrational α and a good sequence S so that μ_{S,α} = ν.
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Dates and versions

hal-03825834 , version 1 (23-10-2022)
hal-03825834 , version 2 (13-11-2023)


  • HAL Id : hal-03825834 , version 2


Emmanuel Lesigne, Anthony Quas, Joseph M Rosenblatt, Maté Wierdl. Generation of measures on the torus with good sequences of integers. 2023. ⟨hal-03825834v2⟩
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