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We study L q -spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we disprove a theorem of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the L q -spectrum. As a further application we provide examples of self-affine measures whose L q -spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the L q -spectra, which in certain cases yield sharp results.

We introduce and study the \emph{Fourier dimension spectrum} which is a continuously parametrised family of dimensions living between the Fourier dimension and the Hausdorff dimension for both sets and measures. We establish some fundamental theory and motivate the concept via several applications, especially to sumset type problems. For example, we study dimensions of convolutions and sumsets, and solve the distance set problem for sets satisfying certain Fourier analytic conditions., Comment: 31 pages, 1 figure

We prove that the Assouad dimension of a parabolic Julia set is $\max\{1,h\}$ where $h$ is the Hausdorff dimension of the Julia set. Since $h$ may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimension are distinct. The box and packing dimensions of the Julia set are known to coincide with $h$ and, moreover, $h$ can be characterised by a topological pressure function. This distinctive behaviour of the Assouad dimension invites further analysis of the Assouad type dimensions, including the Assouad and lower spectra. We compute all of the Assouad type dimensions for parabolic Julia sets and the associated $h$-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2., Comment: 26 pages, 1 figure. This paper used to form part of arXiv:2007.15493 which has subsequently been cut into three pieces. This version contains a new result on Julia sets with Cremer points

We provide a proof of the (well-known) result that the Poincar�� exponent of a non-elementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof only uses elementary hyperbolic and fractal geometry., Expository, 6 pages. This article has been accepted for publication in The American Mathematical Monthly, published by Taylor & Francis

This is a book to be published in 2020 by Cambridge University Press (Tracts in Mathematics Series). It focuses on the Assouad dimension of sets and measures in Euclidean space, as well as many variants on the Assouad dimension, including the lower dimension and Assouad spectrum. It considers these notions in the setting of fractal geometry and topics covered include: weak tangents, self-similar sets, self-affine sets, Mandelbrot percolation, limit sets of Kleinian groups, orthogonal projections, distance sets, and Kakeya sets. It also considers applications of these ideas to other areas, including: embedding theory, number theory, probability theory and functional analysis. It ends with a list of open problems., Comment: Comments welcome. 292 pages, many figures. This material will be published by Cambridge University Press as Assouad Dimension and Fractal Geometry by Jonathan Fraser. This pre-publication version is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. Copyright Jonathan Fraser 2020

Let $p_n$ denote the $n$th prime, and consider the function $1/n \mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that H��lder continuity of this function is equivalent to a parameterised family of Cram��r type estimates on the gaps between successive primes. Here the parameterisation comes from the H��lder exponent. In particular, we show that Cram��r's conjecture is equivalent to the map $1/n \mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n \mapsto 1/n$ is H��lder of all orders but not Lipshitz and this is independent of Cram��r's conjecture., 7 pages, 0 figures

We study the $L^q$-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 $C^{1+\alpha}$ 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 $L^q$-spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function., Comment: 19 Pages, 3 figures

The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret `regularity' in terms of H��lder exponents and establish sharp results for spirals with polynomial winding rates, observing that the sharp H��lder exponent of the forward map and its inverse satisfy a formula reminiscent of Sobolev conjugates. We also investigate the dimension theory of these spirals, in particular, the Assouad dimension, Assouad spectrum and box dimensions. The aim here is to compare the bounds on the H��lder exponents in the winding problem coming directly from knowledge of dimension (and how dimension distorts under H��lder image) with the sharp results. We find that the Assouad spectrum provides the best information, but that even this is not sharp. We also find that the Assouad spectrum is the only `dimension' which distinguishes between spirals with different polynomial winding rates in the superlinear regime., 21 pages, 4 figures

Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in understanding how these different notions fit together, as well as how their subtle differences give rise to different behaviour. Here we survey a new approach in dimension theory, which seeks to unify the study of individual dimensions by viewing them as different facets of the same object. For example, given two notions of dimension, one may be able to define a continuously parameterised family of dimensions which interpolates between them. An understanding of this `interpolation function' therefore contains more information about a given object than the two dimensions considered in isolation. We pay particular attention to two concrete examples of this, namely the \emph{Assouad spectrum}, which interpolates between the box and (quasi-)Assouad dimension, and the \emph{intermediate dimensions}, which interpolate between the Hausdorff and box dimensions., Comment: to appear in Proceedings of Fractal Geometry and Stochastics VI - 18 pages, 6 figures

We study $L^q$-spectra of planar self-affine measures generated by diagonal systems with an emphasis on providing closed form expressions. We answer a question posed by Fraser in 2016 in the negative by proving that a certain natural closed form expression does not generally give the $L^q$-spectrum and, using a similar approach, find counterexamples to a statement of Falconer-Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. In the positive direction we provide new non-trivial closed form bounds in both of the above settings, which in certain cases yield sharp results. We also provide examples of self-affine measures whose $L^q$-spectra exhibit new types of phase transitions. Our examples depend on a combinatorial estimate for the exponential growth of certain split binomial sums., Comment: 27 Pages, 4 Figures

We prove that for an arbitrary upper semi-continuous function $\phi\colon G(1,2) \to [0,1]$ there exists a compact set $F$ in the plane such that $\dim_{\textrm{A}} \pi F = \phi(\pi)$ for all $\pi \in G(1,2)$, where $\pi F$ is the orthogonal projection of $F$ onto the line $\pi$. In particular, this shows that the Assouad dimension of orthogonal projections can take on any finite or countable number of distinct values on a set of projections with positive measure. It was previously known that two distinct values could be achieved with positive measure. Recall that for other standard notions of dimension, such as the Hausdorff, packing, upper or lower box dimension, a single value occurs almost surely., Comment: 14 pages, to appear in Proc. AMS

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 $\geq 2$); 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 Erd\"os-Tur\'an conjecture on arithmetic progressions., Comment: 13 pages, to appear in Bulletin of the London Mathematical Society

We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension $s \in [0,2]$, then the projections have Assouad dimension at least $\min\{1,s\}$ almost surely. Compared to the famous analogue for Hausdorff dimension -- namely \emph{Marstrand's Projection Theorem} -- a striking difference is that the words `at least' cannot be dispensed with: in fact, for many planar self-similar sets of dimension $s < 1$, we prove that the Assouad dimension of projections can attain both values $s$ and $1$ for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the `discrete rotations' case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension, in which case the Assouad dimension is equal to 1. In the `dense rotations' case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no \emph{Falconer's Theorem} for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors., 29 pages, 2 figures, to appear in Proc. Lond. Math. Soc