Schedule for: 24w5264 - On the Interface of Geometric Measure Theory and Harmonic Analysis

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Meet and Greet at the BIRS Lounge (PDC 2nd floor). Guest room key is required to gain access.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Abstract for talk: Let $x$ be a point in the plane. The radial projection $P_x$ maps any point $y \neq x$ in the plane to the unit vector pointing in the direction of $y-x$. We show that for any set $E$ not contained in any line, there exists a point $x \in E$, such that $dim_H P_x(E)= \min\{ 1, \dim_HE\}$. This is an analogue of Marstrand's projection theorem. Joint work with Tuomas Orponen and Pablo Shmerkin.

A set $A\subset\mathbb{Z}$ tiles the integers by translations if there is a set $T\subset\mathbb{Z}$ such that every integer $n\in\mathbb{Z}$ has a unique representation $n=a+t$ with $a\in A$ and $t\in T$. Is there a quick algorithm to determine whether a finite set tiles the integers by translations? Can we estimate the minimal period of such a tiling? What is the relation between tiling and harmonic-analytic properties of a set? We will survey some of the open questions and recent developments in this area. Based in part on my joint work with Benjamin Bruce and Itay Londner.

In this talk we will be interested in fractals which arise naturally in the study of dynamical systems. Until recently, the dimension theory of dynamically defined sets and measures has focused on either the uniformly hyperbolic setting (where the underlying IFS consists of uniform contractions) or the conformal setting (where the underlying IFS consists of conformal maps). In this talk we will survey some recent progress on the dimension theory of sets and measures beyond this scope.

The Fourier spectrum is a continuously parametrised family of dimensions living between the Fourier and Hausdorff dimensions. I will briefly motivate this concept and discuss applications to distance sets and projections.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Canonical Mandelbrot cascades are well-known random measures on the unit interval, exhibiting detailed multifractal structure. The construction relies on a single non-negative random variable $W$ with unit expectation. Under mild assumptions on $W$, we calculate the almost sure Fourier dimension of the cascade measure. We also discuss an analogous problem on the circle $S^1\subset\mathbb{R}^2$. This talk is based on joint work with Changhao Chen and Bing Li.

I will survey some recent progress on the Kakeya conjecture, with a focus on a result of Hong Wang and myself showing that every Kakeya set in $\mathbb{R}^3$ must have Assouad dimension $3$.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Metrical number theory is a fertile meeting ground of many areas of mathematics, including harmonic analysis, geometric measure theory and ergodic theory. This mini-course will present recent developments on a problem about sets of numbers that are partly normal, related to properties of measures supported therein. Each lecture will focus on a distinct feature of the problem, highlighting an idea behind the proof. The course is based on joint work with Junqiang Zhang.

Find notes and slides in these two links:
(notes:) https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/0/26656/files/2024/06/Normal_Rajchman_General_v12.pdf
(slides: ) https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/0/26656/files/2024/06/Normal_Numbers_Background.pdf

These lectures discuss the problem of bounding the number of $\delta$-incidences $\mathcal{I}_{\delta}(P,\mathcal{L}) := \{(p,\ell) \in P \times \mathcal{L} : p \in [\ell]_{\delta}\}$ under various hypotheses on $P \subset \mathbb{R}^{2}$ and $\mathcal{L} \subset \mathcal{A}(2,1)$. The main focus will be on hypotheses relevant for the Furstenberg set problem.

Find abstract and notes here: https://drive.google.com/file/d/1s8qX0ZBPBQjTgO0D8iYeHg2USqWR1Y8a/view
The notes can also be accessed from the page https://sites.google.com/view/tuomaths/teaching

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We study the following problem. Given a modulus of continuity $\omega$, is it true or false that for every function $f$ with modulus of continuity $\omega$, the image of $f$ must have measure zero? This is a joint work with Iqra Altaf.

Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's theorem in which we replace lines with curves.

Favard length of a set is the average length of its orthogonal projections. The Besicovitch projection theorem states the following: for any set E of finite length whose Favard length is positive there exists a Lipschitz graph intersecting E in a set of positive length. In other words, Lipschitz graphs are the only sets of finite length "generating" Favard length. The Favard length problem consists of quantifying this theorem, which is crucial to understand the relation between Favard length and analytic capacity. In this talk I will discuss some recent work on this subject.

Given a Cantor set, does there exist another Cantor set such that the two Cantor sets intersect under all small translations and scalings? The answer is positive for R^1, proven by what we call the containment lemma. In this talk, we will review the containment lemma and generalize it to R^n for uniformly non-degenerate Cantor sets.

Consider a sequence of real numbers increasing to infinity. How large can a subset of the real line get before it is forced to contain some affine image of that sequence? This question fits into a huge body of work in analysis and number theory concerned with constructing large sets that fail to contain prescribed structures. I will discuss recent progress on the above question. Based on joint work with Xiang Gao and Chi Hoi Yip.

The classic Mattila-Sjölin theorem shows that if a compact subset of $\mathbb{R}^d$ has Hausdorff dimension at least $\frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this talk, we present a similar result, namely that if the Hausdorff dimension of a compact subset $E \subset \mathbb{R}^d$, $d \geq 3$, is large enough then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well-known Falconer distance problem which establishes a positive Lebesgue measure for the set of distances instead of it having nonempty interior.

We all work best when our professional communities support our creativity and professional development. In this workshop, we aim to have an open dialogue about a number of talking points meant at creating supportive environments. Find abstract and talking points here: https://docs.google.com/document/d/1hvyKZjjvb-7rPkRVTiikQ3WA0QfARJsgaiG8LpVywYw/edit?usp=sharing

To study the dimension theory of general self-conformal IFSs with overlaps, a widely used method is to study parametrized families of such IFSs instead of individual ones. If such a family satisfies the so-called transversality condition, then it has been known for two decades that for every invariant ergodic measure defined on the symbolic space, there is a full-measure subset of parameters for which the Hausdorff dimension of the push-forward measure can be determined by the ratio entropy over Lyapunov exponent. In this talk, we show that we can choose a full-measure subset of parameters for which the assertion above holds for all ergodic measures simultaneously. We apply this result to the multifractal analysis of typical self-similar measures with overlaps. This is a joint work with Károly Simon and Adam Śpiewak.

Research groups will be formed during the problem session on Monday and rooms will be assigned at that time.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Metrical number theory is a fertile meeting ground of many areas of mathematics, including harmonic analysis, geometric measure theory and ergodic theory. This mini-course will present recent developments on a problem about sets of numbers that are partly normal, related to properties of measures supported therein. Each lecture will focus on a distinct feature of the problem, highlighting an idea behind the proof. The course is based on joint work with Junqiang Zhang.

Find notes and slides in these two links:
(notes:) https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/0/26656/files/2024/06/Normal_Rajchman_General_v12.pdf
(slides: ) https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/0/26656/files/2024/06/Normal_Numbers_Background.pdf

These lectures discuss the problem of bounding the number of $\delta$-incidences $\mathcal{I}_{\delta}(P,\mathcal{L}) := \{(p,\ell) \in P \times \mathcal{L} : p \in [\ell]_{\delta}\}$ under various hypotheses on $P \subset \mathbb{R}^{2}$ and $\mathcal{L} \subset \mathcal{A}(2,1)$. The main focus will be on hypotheses relevant for the Furstenberg set problem.

Find abstract and notes here: https://drive.google.com/file/d/1s8qX0ZBPBQjTgO0D8iYeHg2USqWR1Y8a/view
The notes can also be accessed from the page https://sites.google.com/view/tuomaths/teaching

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Metrical number theory is a fertile meeting ground of many areas of mathematics, including harmonic analysis, geometric measure theory and ergodic theory. This mini-course will present recent developments on a problem about sets of numbers that are partly normal, related to properties of measures supported therein. Each lecture will focus on a distinct feature of the problem, highlighting an idea behind the proof. The course is based on joint work with Junqiang Zhang.

Find notes and slides in these two links:
(notes:) https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/0/26656/files/2024/06/Normal_Rajchman_General_v12.pdf
(slides: ) https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/0/26656/files/2024/06/Normal_Numbers_Background.pdf

These lectures discuss the problem of bounding the number of $\delta$-incidences $\mathcal{I}_{\delta}(P,\mathcal{L}) := \{(p,\ell) \in P \times \mathcal{L} : p \in [\ell]_{\delta}\}$ under various hypotheses on $P \subset \mathbb{R}^{2}$ and $\mathcal{L} \subset \mathcal{A}(2,1)$. The main focus will be on hypotheses relevant for the Furstenberg set problem.

Find abstract and notes here: https://drive.google.com/file/d/1s8qX0ZBPBQjTgO0D8iYeHg2USqWR1Y8a/view
The notes can also be accessed from the page https://sites.google.com/view/tuomaths/teaching

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We are given a self-similar Iterated Function System (IFS) on the real line $$\mathcal{S}:=\{S_1,\dots, S_m\}.$$ We fix a sufficiently large interval $\widehat{I}$ which is sent into itself by all mappings of $\mathcal{S}$. For an arbitrary $n \geq 1$, and $\mathbf{i}=(i_1,\dots,i_n)\in\{1,\dots, m\}^n$ the corresponding level $n$ cylinder interval is $$\begin{equation} \label{2} I_{ i_1,\dots,i_n }: =S_{i_1}\circ\dots\circ S_{i_n}(\widehat{I}). \end{equation}$$ The collection of level $n$ cylinder intervals is $$\mathcal{I}_n := \left\{ I_{ i_1,\dots,i_n }: (i_1,\dots,i_n)\in\{1,\dots, m\}^n \right\}.$$ The attractor is $$\begin{equation} \label{1} \Lambda :=\bigcap _{n=1}^{\infty }\bigcup_{I\in\mathcal{I}_n}I. \end{equation}$$ We say that $\Lambda$ is self-similar set. \bigskip {\bf{Open Problem}} Is there a self-similar set of positive Lebesgue measure and empty interior on the line? \bigskip We consider this problem for randomly perturbed self-similar sets, which are obtained in the following way: In the randomly perturbed case, the $n$-cylinder interval $\widetilde{I}_{ i_1,\dots,i_n }$ corresponding to the indices $\mathbf{i}=(i_1,\dots,i_n)\in\{1,\dots, m\}^n$ is obtained by replacing $S_{i_k}$ in formula $\eqref{2}$ by a random and independent of everything translation $\widetilde{S}_{i_k}$ of $S_{i_k}$ for all $k=1,\dots, n$. Then we build the randomly perturbed attractor in an analogous way to formula $\eqref{1}$ from the randomly perturbed cylinder intervals $\widetilde{I}_{i_1\dots i_n}$. That is $$\widetilde{\mathcal{I}}_n := \left\{ \widetilde{I}_{ i_1,\dots,i_n }: (i_1,\dots,i_n)\in\{1,\dots, m\}^n \right\},$$ and the randomly perturbed self-similar set is $$\widetilde{\Lambda }:= \bigcap _{n=1}^{\infty }\bigcup_{\widetilde{I}\in\widetilde{\mathcal{I}}_n}\widetilde{I}.$$ First, I review results related to the Lebesgue measure and Hausdorff dimension of these randomly perturbed self-similar sets. Then, I turn to our new result (joint with M. Dekking, B. Szekely, and N. Szekeres) about the existence of interior points in these randomly perturbed self-similar sets.

How many rational points with denominator of a given size lie within a certain distance from a compact, "non-degenerate" manifold? We shall discuss a new way of examining the problem: under a combined lens of harmonic analysis (oscillatory integrals) and homogeneous dynamics (quantitative non-divergence estimates). We shall also talk about applications to problems concerning Hausdorff dimension and measure refinements for the set of well-approximable points on non-degenerate manifolds. Based on joint work with Damaris Schindler and Niclas Technau.

Given `n' points inside of a unit square, can we always find 3 points forming a small area triangle? Heilbronn asked for asymptotic upper bounds on the smallest area triangle in an arbitrary set of `n' points. Roth, Komlos, Pintz, and Szemeredi made significant progress on this problem in the late 1970s. We discuss recent work using ideas from fractal geometry to improve on their upper bound for this problem.

We spend our small time sharing with the audience three questions in projection theory. Each of these correspond to the ongoing research program of the presenter. The idea is to give a brief sketch (largely in pictures and examples) of the main questions, with a few comments on the tools utilized to examine each problem. Short problem descriptions are given below. The Favard Length problem (1) asks: if a set in the plane has positive and finite length, but almost-every orthogonal projection of the set has zero length, what is the expected asymptotics of projections of thin neighbourhoods of the set? Continuous Erdős-Beck Theorems (2) assert that, whenever sets are not concentrated (in a dimensional sense) on hyperplanes, then the associated family of spanning lines must have complementarily-large dimension. Falconer-Type Estimates for Dot Products (3) asks one to produce a dimensional threshold for sets in n-dimensional Euclidean space such that the associated dot product set has positive Lebesgue measure. Material for Problem (1) is based on past/ongoing joint work with Izabella Łaba; Material for Problem (2) is based on ongoing joint work with Paige Bright; Material for Problem (3) is based on ongoing joint work with Paige Bright and Steven Senger.

An area of much recent activity in geometric measure theory is the study of conditions on compact sets which guarantee the existence of patterns. A classic example is the Falconer distance problem, which asks how large the Hausdorff dimension must be to ensure a positive measure worth of distances (or, more generally, to ensure the set of distances contains a non-degenerate interval). In this talk I will discuss analogous problems where the notion of Hausdorff dimension is replaced with "thickness", another natural notion of size for Cantor sets.

We derive $L^{3/2}$ estimates for restricted families of projections onto planes and lines in $\mathbb{R}^3$. Such estimates have application in understanding generic intersections of sets in $\mathbb{R}^3$ with light rays and light planes, and imply a sharp $L^{3/2}$ Kakeya maximal inequality for $SL_2$ lines. This is joint work with John Green and Terry Harris.

Research groups will be formed during the problem session on Monday and rooms will be assigned at that time.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss some results and open problems about finite point configurations in large subsets of Euclidean space. This will include joint works with M. Christ, J. Roos and V. Kovac, M. Stipcic.

In this talk, we discuss sharp $L^p$ weighted Fourier restriction estimates of the form $\|E f\|_{L^p(B^{n+1}(0,R),Hdx)}$ $\lessapprox R^{\beta}\|f\|_2$ for a certain range of p, where E is the Fourier extension operator over the truncated paraboloid, and H is a weight function on $\mathbb R^{n+1}$ which is n-dimensional up to scale $\sqrt R$. Joint work with Jianhui Li, Hong Wang, and Ruixiang Zhang.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

In the last decade it was realized that many classical problems in geometric measure theory are more tractable for Ahlfors regular sets (perhaps in an approximate or finitary sense) than for general sets. I will survey some recent progress in this direction, obtained in (separate) joint work with H. Wang and with T. Orponen. The main goal of the talk will be to indicate how the Ahlfors regularity assumption comes into play.

We will discuss Falconer's distance problem in a class of non-Euclidean Banach spaces. This is joint work with Iqra Altaf and Ryan Bushling.