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We consider the ensemble of curves {
, : ∈(0, 1], ∈ℕ} obtained by linearly interpolating the values of the normalized theta sum −1/2∑=0−1exp(πi2), 0≤<. We prove the existence of limiting finite-dimensional distributions for such curves as →∞, when is distributed according to any probability measure , absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [
(1999) 127–153] and Jurkat and van Horne [
...
We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.
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