Packing dimension of mean porous measures

We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A\subset\mathbb R$.Comment: Revised versio

Minkowski dimension for measures

We introduce a Minkowski dimension for measures and show that it can be used to characterise the Minkowski dimension of a compact metric space. We also study its relationship with other concepts in dimension theory.Comment: 1 figure; the proof of Theorem 3.3 in v1 contains a gap; we improved the main result and polished the presentatio

The Dutch Healthy Diet Index : development, evaluation, and application

The Dutch Healthy Diet index – Development, Evaluation, and Application Linde van Lee Abstract Background: Dietary indices evaluate the conformity of an individual’s diet with pre-defined standards. Generally, dietary guidelines are used for this purpose. As no index based on the current dietary guidelines was available in the Netherlands, the aim of the present thesis was to develop, evaluate, and apply a dietary index for use in the country. Methods and results: The Dutch Healthy Diet index (DHD-index) was developed on the basis of the 2006 Dutch dietary guidelines using data relating to 749 young adults who completed two 24-hour recalls in the Dutch national food consumption survey 2003. The index comprises ten components on physical activity, vegetables, fruit, dietary fibre, saturated fatty acids, trans fatty acids, consumption occasions with acidic drinks and foods, sodium, and alcohol. Scores for each component range between 0 (no adherence) and 10 (complete adherence) points. The DHD-index was inversely associated with energy intake and positively associated with most micronutrient intakes when adjusted for energy intake. We compared the DHD-index score based on two 24-hour recalls with the index based on the food frequency questionnaires (FFQ) of 121 adults from the European Food Consumption Validation study. We revealed an acceptable correlation (r=0.48) and absolute agreement between the indices based on the two methods. The prospective relationship with mortality outcomes was studied in 3593 of the Rotterdam Study participants who were followed for 20 years. The DHD-index per 10 points increment was associated with a 9% (95% CI 0.87-0.96) risk reduction for all-cause mortality, and non-significantly associated with risk reductions for cardiovascular disease, coronary heart disease, and stroke mortality. Among women, shared dinners were associated with lower DHD-index scores for that day than solo dinners in 1740 participants who contributed multiple 24-hour recalls in the Nutrition Questionnaires plus study. Among men and women, dinners shared with family members were associated with a higher DHD-index score on that day than dinners shared with others. Furthermore, in a subsample of 1235 participants in the Nutrition Questionnaires plus study, we evaluated the DHD-index based on the newly developed 34-item DHD-FFQ, a short questionnaire to assess diet quality in time-limited settings. The DHD-index based on the DHD-FFQ showed an acceptable correlation (r=0.56) with the index based on a 180-item FFQ, but showed a large variation in bias at individual level. Conclusions: The DHD-index based on an FFQ, on multiple 24-hour recalls, or on the DHD-FFQ was considered a valid tool to rank participants according to their diet quality. The DHD-index was therefore considered useful to monitor populations, study diet–disease associations, and identify subpopulations at risk of poor diet quality.</p

The Hausdorff and dynamical dimensions of self-affine sponges : a dimension gap result

We construct a self-affine sponge in R 3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space

Lyapunov spectrum of asymptotically sub-additive potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map that is not upper-semi continuous.Comment: 44 page

Distance sets, orthogonal projections, and passing to weak tangents

The author is supported by a Leverhulme Trust Research Fellowship (RF-2016-500).We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.PostprintPeer reviewe

Asymptotically sharp dimension estimates for kk-porous sets

In ${\mathsf R}^n$, we establish an asymptotically sharp upper bound for the upper Minkowski dimension of $k$-porous sets having holes of certain size near every point in $k$ orthogonal directions at all small scales. This bound tends to $n-k$ as $k$-porosity tends to its maximum value