The number of studies examining associations between training load and injury has increased exponentially. As a result, many new measures of exposure and training-load-based prognostic factors have been created. The acute:chronic workload ratio (ACWR) is the most popular. However, when recommending the manipulation of a prognostic factor in order to alter the likelihood of an event, one assumes a causal effect. This introduces a series of additional conceptual and methodological considerations that are problematic and should be considered. Because no studies have even tried to estimate causal effects properly, manipulating ACWR in practical settings in order to change injury rates remains a conjecture and an overinterpretation of the available data. Furthermore, there are known issues with the use of ratio data and unrecognized assumptions that negatively affect the ACWR metric for use as a causal prognostic factor. ACWR use in practical settings can lead to inappropriate recommendations, because its causal relation to injury has not been established, it is an inaccurate metric (failing to normalize the numerator by the denominator even when uncoupled), it has a lack of background rationale to support its causal role, it is an ambiguous metric, and it is not consistently and unidirectionally related to injury risk. Conclusion: There is no evidence supporting the use of ACWR in training-load-management systems or for training recommendations aimed at reducing injury risk. The statistical properties of the ratio make the ACWR an inaccurate metric and complicate its interpretation for practical applications. In addition, it adds noise and creates statistical artifacts.

We introduce a continuum of dimensions which are 'intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that |U | ≤ |V | θ for all sets U, V used in a particular cover, where θ ∈ [0, 1] is a parameter. Thus, when θ = 1 only covers using sets of the same size are allowable, and we recover the box dimensions, and when θ = 0 there are no restrictions, and we recover Hausdorff dimension.We investigate many properties of the intermediate dimension (as a function of θ), including proving that it is continuous on (0, 1] but not necessarily continuous at 0, as well as establishing appropriate analogues of the mass distribution principle, Frostman's lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford-McMullen carpets.Mathematics Subject Classification 2010 : primary: 28A80; secondary: 37C45.

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We study the typical growth rate of the number of words of length n which can be extended to β-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter β is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.