We study the descriptive complexity of sets of points defined by restricting the statistical behaviour of their orbits in dynamical systems on Polish spaces. Particular examples of such sets are the sets of generic points of invariant Borel probability measures, but we also consider much more general sets (for example, $\alpha $-Birkhoff regular sets and the irregular set appearing in the multifractal analysis of ergodic averages of a continuous real-valued function). We show that many of these sets are Borel in general, and all these are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is, in some cases, necessary to obtain Borelness. When these sets are Borel, we measure their descriptive complexity using the Borel hierarchy. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that these sets are properly located at the third level of the hierarchy for many dynamical systems. To demonstrate that the specification property is a sufficient, but not necessary, condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is $\boldsymbol {\Pi }^0_3$-complete.

High-quality evidence from prospective longitudinal studies in humans is essential to testing hypotheses related to the developmental origins of health and disease. In this paper, the authors draw upon their own experiences leading birth cohorts with longitudinal follow-up into adulthood to describe specific challenges and lessons learned. Challenges are substantial and grow over time. Long-term funding is essential for study operations and critical to retaining study staff, who develop relationships with participants and hold important institutional knowledge and technical skill sets. To maintain contact, we recommend that cohorts apply multiple strategies for tracking and obtain as much high-quality contact information as possible before the child’s 18th birthday. To maximize engagement, we suggest that cohorts offer flexibility in visit timing, length, location, frequency, and type. Data collection may entail multiple modalities, even at a single collection timepoint, including measures that are self-reported, research-measured, and administrative with a mix of remote and in-person collection. Many topics highly relevant for adolescent and young adult health and well-being are considered to be private in nature, and their assessment requires sensitivity. To motivate ongoing participation, cohorts must work to understand participant barriers and motivators, share scientific findings, and provide appropriate compensation for participation. It is essential for cohorts to strive for broad representation including individuals from higher risk populations, not only among the participants but also the staff. Successful longitudinal follow-up of a study population ultimately requires flexibility, adaptability, appropriate incentives, and opportunities for feedback from participants.

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

A Cantor series expansion for a real number x with respect to a basic sequence $Q=(q_1,q_2,\dots )$, where $q_i \geq 2$, is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a $\boldsymbol {\Pi }^0_3$-complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is $\boldsymbol {\Pi }^0_3$-complete, and if Q is $1$-divergent then the sets of normal and ratio normal numbers are $\boldsymbol {\Pi }^0_3$-complete. We further show that all five non-trivial differences of these sets are $D_2(\boldsymbol {\Pi }^0_3)$-complete if $\lim _i q_i=\infty $ and Q is $1$-divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.

The scientific community and most mainstream agriculturalists typically design fertilizer recommendations to provide a ‘sufficient level of available nutrients’ to meet the annual N, P and K requirements of common field crops. Soil balancing is another approach to managing soil fertility that focuses on the levels of Ca, Mg and K to achieve a desired base cation saturation ratio (BCSR). Soil balancing is believed to be practiced frequently by organic and other alternative farmers but is viewed skeptically by conventional agricultural scientists due to a lack of support for the idea in the published scientific literature. This study represents a pioneering effort to collect systematic data on the extent of soil balancing, how it is practiced and the types of outcomes reported by organic farmers. Our survey of over 850 farmers who grow certified organic corn in Indiana, Michigan, Ohio and Pennsylvania found that over half report using a soil-balancing approach based on BCSR. Their practice of soil balancing frequently includes more than management of base cations, but also uses a wide range of soil amendment products (such as purchased organic NPK fertilizers, micronutrients, microbial stimulants and soil inoculants) other than those applied specifically for cation balance. Farms that rely on vegetable and dairy production for most of their income, and Amish farmers who rely on horses for fieldwork, were more likely to report using a soil-balancing program. Self-described soil balancers perceived positive agronomic outcomes from the use of a BCSR program, including improvements in soil physical and biological properties and improved crop health and quality. Although farmers in our study report extensive use and positive perceived outcomes from soil-balancing methods, the scientific research literature has been unable to reproduce evidence that manipulating soil base cation levels has any systematic effect on crop yield. Future research could consider the interacting effects of BCSR with other field management practices to more closely approximate the actual practices of farmers.

Let $\mathcal {N}(b)$ be the set of real numbers that are normal to base b. A well-known result of Ki and Linton [19] is that $\mathcal {N}(b)$ is $\boldsymbol {\Pi }^0_3$-complete. We show that the set ${\mathcal {N}}^\perp (b)$ of reals, which preserve $\mathcal {N}(b)$ under addition, is also $\boldsymbol {\Pi }^0_3$-complete. We use the characterization of ${\mathcal {N}}^\perp (b),$ given by Rauzy, in terms of an entropy-like quantity called the noise. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of ${\mathcal {N}}^\perp (b)$. We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $\boldsymbol {\Pi }^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

Assuming AD, we show that all of the ordinals below $\delta _5^1$ represented by descriptions (c.f. [2], but also defined below) are cardinals. Using this analysis we also get a simple representation for the cardinal structure below $\delta _5^1$ . As an application, we compute the cofinalitites of all cardinals below $\delta _5^1$ .

We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal κ < ℵ ε0 in a recursive definition of a canonical measure assignment. We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals).