Dedicated to the memory of Bernd O. Stratmann -A good friend, colleague and mentor.Abstract. Given an α > 1 and a θ with unbounded continued fraction entries, we characterise new relations between Sturmian subshifts with slope θ with respect to (i) an α-Höder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of θ, and (iii) complexity notions which we call α-repetitive, α-repulsive and α-finite; generalisations of the properties known as linearly repetitive, repulsive and power free, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is 2/(α + 1).

At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely α-repetitive, α-repulsive and α-finite (α ≥ 1), have been introduced and studied. We establish the equivalence of α-repulsive and α-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite 2-group G. In particular, we show that these subshifts provide examples that demonstrate α-repulsive (and hence α-finite) is not equivalent to α-repetitive, for α > 1. We also give necessary and sufficient conditions for these subshifts to be α-repetitive, and α-repulsive (and hence α-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.

Letting T denote an ergodic transformation of the unit interval and letting f : [0, 1) → R denote an observable, we construct the f -weighted return time measure µ y for a reference point y ∈ [0, 1) as the weighted Dirac comb with support in Z and weights f • T z (y) at z ∈ Z, and if T is non-invertible, then we set the weights equal to zero for all z < 0. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of µ y consists of a trivial atom and an absolutely continuous part, almost surely with respect to y. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider the family of rigid rotations T α : x → x + α mod 1 with rotation number α ∈ R + . In contrast to when T is mixing, we observe that the diffraction of µ y is pure point, almost surely with respect to y. Moreover, if α is irrational and the observable f is Riemann integrable, then the diffraction of µ y is independent of y. Finally, for a converging sequence (α i ) i∈N of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.Here, for z ∈ Z, we let δ z denote the Dirac point mass at z. In this note we answer the following questions.