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We study the Lq-spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are C1+α and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the Lq-spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function. [ABSTRACT FROM AUTHOR]

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely 'as large as possible' (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments. [ABSTRACT FROM AUTHOR]

Let pn denote the n th prime, and consider the function 1/n ↦ 1 / pn which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parametrised family of Cramér type estimates on the gaps between successive primes. Here the parametrisation comes from the Hölder exponent. In particular, we show that Cramér's conjecture is equivalent to the map 1 / n ↦ 1 / p n being Lipschitz. On the other hand, we show that the inverse map 1 / p n ↦ 1 / n is Hölder of all orders but not Lipschitz and this is independent of Cramér's conjecture. [ABSTRACT FROM AUTHOR]

We consider Schmidt’s game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behavior of objects in a metric space, mostly in the context of fractal dimensions, and the notion of “generic” we adopt is that of being winning for Schmidt’s game.We find properties whose corresponding sets are winning for Schmidt’s game that are starkly different from previously established, and well-known, properties which are generic in other contexts, such as being residual or of full measure. [ABSTRACT FROM AUTHOR]

The aims of this study were to examine the effects of task and environmental constraints on axial synergies and to find an association between synergies and arm acceleration as a performance variable. Participants of this study were 10 expert tennis players (age: 34.4 ± 7.46) who voluntarily took part and executed 60 serves under two different conditions: no-opponent and opponent. An inertial motion unit (IMU) capture system was used to calculate the 3D angular joint motions in the neck, back and lumbar segments. The results of the principal component analysis showed that the redundancy in the axial segments is decomposed into 2 main synergies that are responsible for the loading (backward swing) and firing phase (forward swing). The total variance and loading synergy variance were significantly lower in the topspin service than other service types. The emerged firing synergy was strongly associated with the arm acceleration regardless of service type. In conclusion, the effective strategy to utilise the axial motions in the trunk is through creating functional synergies that have a flexible role based on the type of service and conditions. The topspin service showed less coordination variability relative to other types of service and serving in the opponent condition required participants to change the nature of synergy among the axial segments. These findings support the design of practice that emphasises the importance of more realistic contexts with special attention given to the order of different service types. [ABSTRACT FROM AUTHOR]

We study equilibrium measures (Käenmäki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of a coordinate projection of the measure. In particular, we do this by showing that the Käenmäki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type. [ABSTRACT FROM AUTHOR]

We provide estimates for the dimensions of sets in R which uniformly avoid finite arithmetic progressions (APs). More precisely, we say F uniformly avoids APs of length k ≥ 3 if there is an ε > 0 such that one cannot find an AP of length k and gap length Δ > 0 inside the εΔ neighbourhood of F⁠. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of k and ε⁠. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a "reverse Kakeya problem:" we show that if the dimension of a set in Rd is sufficiently large, then it closely approximates APs in every direction. [ABSTRACT FROM AUTHOR]

The aims of this study were to examine the effect of different environmental constraints on kinematic multi-segment coordination patterns during the service and its coordination with service time variability. Ten expert tennis players (Age: 34.1 ± 5.3) volunteered to take part in this study. Participants served 30 times in 3 different conditions: control, target and opposition. The order of conditions was counterbalanced between participants. A wireless 3D motion capture system (STT Co, Spain) was used to measure 7 joint motions, with a 17 degrees of freedom biomechanical model created to capture the entire service action. Results of the principal component analysis showed that 4 synergies were created; however, their roles were changed relative to the perception of the environment. The results of repeated-measures analysis of variance did not show any significant difference on total variance and individual principal components between conditions; however, one synergy pattern significantly predicted the service time variability in both control and opposition conditions. In conclusion, the findings demonstrated that expert performers reduce the joint dimensionality by creating functional synergies in different phases of service and adapt the service action according to the perception of the environment. [ABSTRACT FROM AUTHOR]

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this 'expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $\text{SO}(d)$ for $d\geq 3$. We also obtain new upper bounds, derived using sumsets, for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the weak separation property is satisfied, that is, the overlaps are controllable, then the 'expected formula' does hold. [ABSTRACT FROM AUTHOR]

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension. [ABSTRACT FROM PUBLISHER]

Abstract: We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including the presence (or lack of) arithmetic progressions (or patches in dimensions at least 2), the structure of tangent sets, and the Assouad dimension. We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non‐empty interior and to ‘asymptotically’ containing arbitrarily large arithmetic patches. We present some applications of our results concerning sets of integers, which include a weak solution to the Erdös–Turán conjecture on arithmetic progressions. [ABSTRACT FROM AUTHOR]

We study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets. [ABSTRACT FROM AUTHOR]