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We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $\mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture., Comment: 43 pages, no figures

We make progress on several interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, exceptional estimates for orthogonal projections, and the dimension of Furstenberg sets. We give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0 < {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A < 1$ and ${\dim_{\mathrm{H}}} B + {\dim_{\mathrm{H}}} C > {\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $${\dim_{\mathrm{H}}} (A + cB) > {\dim_{\mathrm{H}}} A. $$ Here we only mention special cases of our results on projections and Furstenberg sets. We prove that every $s$-Furstenberg set $F \subset \mathbb{R}^{2}$ has Hausdorff dimension $$ {\dim_{\mathrm{H}}} F \geq \max\{ 2s + (1 - s)^{2}/(2 - s), 1+s\}.$$ We prove that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ associated with a $t$-Ahlfors-regular line set has $${\dim_{\mathrm{H}}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$ Let $\pi_{\theta}$ denote projection onto the line spanned by $\theta\in S^1$. We prove that if $K \subset \mathbb{R}^{2}$ is a Borel set with ${\dim_{\mathrm{H}}}(K)\le 1$, then $$ {\dim_{\mathrm{H}}} \{\theta \in S^{1} : {\dim_{\mathrm{H}}} \pi_{\theta}(K) < u\} \leq \max\{ 2(2u - {\dim_{\mathrm{H}}} K),0\}, $$ whenever $u \leq {\dim_{\mathrm{H}}} K$, and the factor "$2$" on the right-hand side can be omitted if $K$ is Ahlfors-regular., Comment: 73 pages. v4: improved Theorem 5.61 and Remark 5.67

We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+\epsilon$, where $\epsilon>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let $X,Y \subset \mathbb{R}^{2}$ be non-empty Borel sets. If $X$ is not contained on any line, we prove that \[ \sup_{x \in X} \dim_{\mathrm{H}} \pi_{x}(Y) \geq \min\{\dim_{\mathrm{H}} X,\dim_{\mathrm{H}} Y,1\}. \] If $\dim_{\mathrm{H}} Y > 1$, we have the following improved lower bound: \[ \sup_{x \in X} \dim_{\mathrm{H}} \pi_{x}(Y \, \setminus \, \{x\}) \geq \min\{\dim_{\mathrm{H}} X + \dim_{\mathrm{H}} Y - 1,1\}. \] Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if $X \subset \mathbb{R}^{2}$ is a Borel set with the property that $\dim_{\mathrm{H}} (X \, \setminus \, \ell) = \dim_{\mathrm{H}} X$ for all lines $\ell \subset \mathbb{R}^{2}$, then the line set spanned by $X$ has Hausdorff dimension at least $\min\{2\dim_{\mathrm{H}} X,2\}$. While the results above concern $\mathbb{R}^{2}$, we also derive some counterparts in $\mathbb{R}^{d}$ by means of integralgeometric considerations. The proofs are based on an $\epsilon$-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren., Comment: 31 pages. This paper supersedes arXiv:2205.13890

We prove two new exceptional set estimates for radial projections in the plane. If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K > 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} \pi_{x}(K) \leq \sigma\} \leq \max\{1 + \sigma - \dim_{\mathrm{H}} K,0\}, \qquad \sigma \in [0,1).$$ If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K \leq 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} \pi_{x}(K) < \dim_{\mathrm{H}} K\} \leq 1.$$ The finite field counterparts of both results above were recently proven by Lund, Thang, and Huong Thu. Our results resolve the planar cases of conjectures of Lund-Thang-Huong Thu, and Liu., Comment: 25 pages

We establish the dimension version of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension $d/2$; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension $1$ has Hausdorff dimension at least $(\sqrt{5}-1)/2\approx 0.618$. In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension $d/2$. These results rely on new estimates for the dimensions of radial projections that may have independent interest., Comment: v3: Many small corrections, incorporates referees suggestions. 71 pages, 4 figures. To appear in GAFA

We survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing $\times 2$, $\times 3$ conjecture of H. Furstenberg in ergodic theory, and the distance set problem in geometric measure theory introduced by K. Falconer. We discuss some of the ideas behind our solution of Furstenberg's slicing conjecture, and recent progress in Falconer's problem. While these two problems are on the surface rather different, we emphasize some common themes in our approach: analyzing fractals through a combinatorial description in terms of ``branching numbers'', and viewing the problems through a ``multiscale projection'' lens., Comment: 25 pages, submitted to the Proceedings ofthe ICM 2022

Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$, and $t \in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in $\mathbb{R}^{2}$ is no smaller than $2s + \epsilon$, where $\epsilon > 0$ depends only on $s$ and $t$. For $s>1/2$ and $t = 1$, this is an $\epsilon$-improvement over a result of Wolff from 1999. The same method also yields an $\epsilon$-improvement to Kaufman's projection theorem from 1968. We show that if $s \in (0,1)$, $t \in (s,2]$ and $K \subset \mathbb{R}^{2}$ is an analytic set with $\dim_{\mathrm{H}} K = t$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{H}} \pi_{e}(K) \leq s\} \leq s - \epsilon,$$ where $\epsilon > 0$ only depends on $s$ and $t$. Here $\pi_{e}$ is the orthogonal projection to $\mathrm{span}(e)$., Comment: 64 pages, 2 figures. v3: corrected and improved Proposition 5.2. To appear in Duke Math J

We prove that if $\alpha\in (0,1/2]$, then the packing dimension of a set $E\subset\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\ge \alpha$ is at least $1/2+\alpha+c(\alpha)$ for some $c(\alpha)>0$. In particular, this holds for $\alpha$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $\ge\alpha$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $\alpha$-Furstenberg sets over the "trivial" estimate for all values of $\alpha\in (0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $\alpha$-Furstenberg set resembles a set of dimension close to $1/2+\alpha$, if they exist, are rather sparse., Comment: 12 pages. v2: incorporates referee's comments, to appear in Journal d'Analyse Math\'ematique

We show that non-trivial $\times N$-invariant sets in $[0,1]^d$, such as the Sierpi\'{n}ski carpet and the Sierpi\'{n}ski sponge, are tube-null, that is, they can be covered by a union of tubular neighbourhoods of lines of arbitrarily small total volume. This introduces a new class of tube-null sets of dimension strictly between $d-1$ and $d$. We utilize ergodic-theoretic methods to decompose the set into finitely many parts, each of which projects onto a set of Hausdorff dimension less than $1$ in some direction. We also discuss coverings by tubes for other self-similar sets, and present various applications., Comment: 24 pages, 2 figures

We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to $1$ in $\mathbb{R}^2$ or close to $3/2$ in $\mathbb{R}^3$, then for $y\in A$ outside of a very sparse set, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff dimension at least $1/2+c$, where $c$ is universal. Furthermore, the same holds if the distances are taken with respect to a $C^2$ norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between $\delta$-balls and $\delta$-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into ``Frostman pieces'' that may be of independent interest., Comment: 51 pages. v2: several fixes and clarifications, main results unchanged but numbering has changed

We show that given $\alpha \in (0, 1)$ there is a constant $c=c(\alpha) > 0$ such that any planar $(\alpha, 2\alpha)$-Furstenberg set has Hausdorff dimension at least $2\alpha + c$. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps., Comment: 29 pages. v2: many small corrections, results unchanged

We investigate the box dimensions of compact sets in $\mathbb{R}^2$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least $\frac{4}{7}$ and can be as low as $\frac{2}{3}$. This quantifies in a certain sense how far the unit circle is from being a difference set., Comment: 13 pages, 2 figures. v3: the proof of lower bound in dimension d\ge 3 contained a gap hence we have removed this claim; the lower bounds in the plane remain valid

We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of $F \subseteq \rn$ is no greater than $m$, then the box and packing dimensions of $F$ are preserved under orthogonal projections onto almost all $m$-dimensional subspaces. We also show that the threshold $m$ for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian., Comment: 10 pages, 1 figure. To appear in Journal of Fractal Geometry

We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $\varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemer\'{e}di bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent to box or Assouad dimension for this problem, and obtain a lower bound for Fourier dimension., Comment: 14 pages. v3: minor corrections and clarifications

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff dimension at least $37/54\approx 0.685$. Moreover, if $s\in (1,3/2]$, then outside of a set of exceptional $y$ of Hausdorff dimension at most $1$, the pinned distance set $\{ |x-y|:x\in A\}$ has Hausdorff dimension $\ge \tfrac{2}{3}s$ and packing dimension at least $ \tfrac{1}{4}(1+s+\sqrt{3s(2-s)}) \ge 0.933$. These estimates improve upon the existing ones by Bourgain, Wolff, Peres-Schlag and Iosevich-Liu for sets of Hausdorff dimension $>1$. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions., Comment: 60 pages, 2 figures. Incorporates referee comments. To appear in GAFA

We prove that if $A$ is a Borel set in the plane of equal Hausdorff and packing dimension $s>1$, then the set of pinned distances $\{ |x-y|:y\in A\}$ has full Hausdorff dimension for all $x$ outside of a set of Hausdorff dimension $1$ (in particular, for many $x\in A$). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint $s=1$., Comment: 19 pages, no figures. v2: minor corrections, all results unchanged

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{\eta_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is H\"older continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. {\L}aba in connection to the restriction problem for fractal measures., Comment: 96 pages, 5 figures. v4: The definition of the metric changed in Section 8. Polishing notation and other small changes. All main results unchanged

We study the relationship between the sizes of two sets $B, S\subset\mathbb{R}^2$ when $B$ contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of $S$, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set $B$ of Hausdorff dimension $1$ which contains the boundary of an axes-parallel square with center in every point $[0,1]^2$, but prove that such a $B$ must have packing and lower box dimension at least $\tfrac{7}{4}$, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others., Comment: 20 pages, no figures