arXiv subjects

Krystal Taylor

Publications and source records attributed to Krystal Taylor.

At least 19 recordsLinked to original sources

arXiv2026

Minkowski sums with convex curves without pointwise Fourier decay

Let $\Gamma\subset\mathbb R^2$ be a compact convex graph and define \[ T(\Gamma) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+\Gamma|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(\Gamma)=1$. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention $\widehat\nu(\xi)=\int e^{-2\pi i x\cdot\xi}\,d\nu(x)$. We construct a strictly convex Lipschitz graph $\Gamma$ with $T(\Gamma)=1$ such that, for every nontrivial subarc $\Gamma_0$ and every $\alpha>0$, \[ \limsup_{|\xi|\to\infty} |\xi|^\alpha \left| \widehat{H^1|_{\Gamma_0}}(\xi) \right|= \infty. \] We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if $\Gamma$ contains a positive-length subset of a $C^2$ curve whose curvature is bounded away from zero, then $|E+\Gamma|>0$ whenever $\dim_{\mathrm H}(E)>1$. For a nondegenerate graph this gives $T(\Gamma)=1$. For convex graphs it implies, in particular, that $T(\Gamma)=1$ whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold $T(\Gamma)=1$.

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arXiv2026

Arithmetic-progression gap sets in Cantor sets

We address the question of which common differences can arise in arithmetic progressions contained in fractal sets. For a compact set $C\subset\mathbb R$, we investigate not only whether arithmetic progressions occur in $C$, but the full collection of their common differences. More generally, for a finite pattern $P$, we study the set of scales at which affine copies of $P$ appear in $C$. For affine self-similar sets satisfying strong separation, we obtain explicit restrictions on admissible common differences. Specializing to middle-$\varepsilon$ Cantor sets, we prove that the longest arithmetic progression has length four whenever $3-2\sqrt2<\varepsilon\le 1/3$, showing that the maximal progression length drops immediately from six at the critical parameter $\varepsilon=3-2\sqrt2$. We further develop recursive bounds for the sets of admissible common differences and derive explicit blackout intervals, namely ranges of scales for which arithmetic progressions cannot occur. On the positive side, sufficiently thick Cantor sets exhibit the opposite behavior. Combining a refinement of the Hunt-Kan-Yorke construction with the Newhouse Gap Lemma, we prove that every sufficiently small common difference occurs in a three-term arithmetic progression. In particular, if the largest bounded gap of a Cantor set is at most $0.067 diam(C)$ and its thickness is at least $6.96268\ldots$, then every common difference in $(0,0.435 diam(C)]$ occurs in a three-term arithmetic progression contained in $C$. Analogous interval results are obtained for four-term arithmetic progressions and asymmetric three-point patterns.

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arXiv2026

Applications of Nonlinear Projections to Rectifiable 1-sets

Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $\mathbb{R}^d$, a classical theorem of Federer shows that the 1-dimensional Hausdorff measure of such sets is controlled by the multiplicity-weighted lengths of finitely many linearly independent projections. We develop a framework for extending Federer's result into a diverse set of nonlinear problems. This technique yields a unified approach for studying sets through their lower-dimensional nonlinear images, as well as studying the exceptional sets which exhibit poor projective behavior. As illustrations of our technique, we show that (i) every 1-rectifiable set contains a pin whose pinned distance set has positive Lebesgue measure, and that the exceptional set of pins for which this fails is contained in a $(d-2)$-dimensional affine subspace; (ii) planar radial projections of a 1-rectifiable set can fail to have positive length from at most one vantage point unless the set is essentially linear; and finally (iii) unions of circles centered on a 1-rectifiable set have positive area under mild assumptions on the radius function.

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arXiv2026

Favard length and generalized projections

We investigate generalized Favard lengths associated to smooth families of nonlinear projections. Under suitable regularity and transversality assumptions, we prove that generalized projections are locally comparable to orthogonal projections on sufficiently small scales. This yields a comparison principle that transfers quantitative upper bounds for classical Favard length to broad classes of nonlinear projection families. As a consequence, known upper bounds for the Favard length of purely unrectifiable self-similar 1-sets yield corresponding upper bounds for their generalized Favard lengths. We also prove that the union of circles with centers in a purely unrectifiable self-similar 1-set has Lebesgue measure zero whenever the radii vary sufficiently slowly. More generally, the same method yields measure estimates for unions of curves arising from suitable level-set families.

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arXiv2026

A Quantified Two-projection Theorem for Nonlinear Projections

The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely $1$-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. Further, we use a multiscale framework to obtain a quantitative version of our nonlinear two-projection theorem. Our arguments utilize methods introduced by Tao, who provided a quantitative treatment of the classic linear two-projection theorem.

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arXiv2026

Positive Measure of Unions of Variable Surfaces

Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $\phi(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{\phi(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$ \left|\bigcup_{x \in E} \{y : \phi(x,y)=1\}\right|>0. $$ This extends the positivity theorem of Mitsis ($d\geq3$) and Wolff ($d=2$) for spheres to a general variable coefficient setting via $L^2$ estimates for Fourier integral operators. The argument also shows that positivity is stable under finite-order degeneracies of the Monge--Amp\`ere determinant through the weighted averaging theory of Sogge and Stein. We next consider variable level sets $$ \Sigma_x=\{y:\phi(x,y)=t(x)\}, $$ where $t(x)$ is measurable. A maximal operator argument yields positivity under the condition $\dim_{\mathcal H}(E)>2$. We show that this loss reflects a genuine geometric obstruction related to Kakeya-type compression phenomena. In contrast, under a direct geometric intersection hypothesis controlling overlaps of the hypersurfaces $\Sigma_x$, we recover the full threshold $\dim_{\mathcal H}(E)>1$ for arbitrary measurable selections $t=t(x)$. At the endpoint $\dim_{\mathcal H}(E)=1$, we obtain positivity under the additional assumption that $E$ is $1$-rectifiable with $\mathcal H^1(E)>0$. We also show that positivity of Lebesgue measure does not in general imply interior regularity: even for large or rectifiable parameter sets, the resulting unions may have empty interior. Finally, we discuss extensions to higher co-dimension families and the role of geometric structure in preventing compression phenomena.

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arXiv2025

Interior of distance trees over thin Cantor sets

It is known that if a compact set $E$ in $\mathbb{R}^d$ has Hausdorff dimension greater than $(d+1)/2$, then its $n$-chain distance set $$\Delta^n(E) = \left\{\left(\left|x^1-x^2\right|,\cdots, \left|x^{n}- x^{n+1}\right|\right)\in \mathbb{R}^n: x^i \in E, x^i\neq x^j \text{ for } i\neq j \right\}$$ has nonempty interior for any $n\in \mathbb{N}$. In this paper, we prove that for every Cantor set $K\subset \mathbb{R}^d$ and for every $n\in\mathbb{N}$, there exists $\widetilde{K}\subset \mathbb{R}^d$ such that the pinned $n$-chain distance set of $K\times \widetilde{K}\subset \mathbb{R}^{2d}$ has nonempty interior, and hence, that $\Delta^n(K\times \widetilde{K})$ has nonempty interior. Our results do not depend on the Newhouse gap lemma but rather on the containment lemma recently introduced by Jung and Lai. Our results generalize three-fold: to arbitrary finite trees, to higher dimensions, and to maps that have non-vanishing partials. As an application, we provide a class of examples of Cantor sets $E\subset \mathbb{R}^{2d}$ so that for any $s\geq d$, $\dim_{\rm H}(E)= s$ and $\Delta_x^n(E)^\circ{}\neq \varnothing$ for some $x\in E$.

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arXiv2025

Triangles in the Plane and arithmetic progressions in thick compact subsets of $\mathbb{R}^d$

This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb{R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb{R}^d$ contains a similar copy of any linear $3$-point configuration (such as a $3$-point arithmetic progression) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition for $d\geq 2$; or, when $d=1$, the result holds provided the Newhouse thickness of $A$ is at least $1$. Moreover, we prove that compact sets $A\subset \mathbb{R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of $C$ is at least $1$. These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.These are among the first results in the literature to give explicit criteria for the occurrence of three-point configurations in the plane.

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arXiv2025

Point configurations in sets of sufficient topological structure and a topological {E}rd\H{o}s similarity conjecture

We explore the occurrence of point configurations within non-meager (second category) Baire sets. A celebrated result of Steinhaus asserts that $A+B$ and $A-B$ contain an interval whenever $A$ and $B$ are sets of positive Lebesgue measure in $\mathbb{R}^n$ for $n\geq 1$. A topological analogue attributed to Piccard asserts that both $AB$ and $AB^{-1}$ contain an interval when $A,B$ are non-meager (second category) Baire sets in a topological group. We explore generalizations of Piccard's result to more complex point configurations and more abstract spaces. In the Euclidean setting, we show that if $A\subset \mathbb{R}^d$ is a non-meager Baire set and $F=\{x_n\}_{n\in\mathbb{N}}$ is a bounded sequence, then there is an interval of scalings $t$ for which $tF+z\subset A$ for some $z\in \mathbb{R}^d$. That is, the set $$\Delta_F(A)=\{t\in\mathbb{R}: \exists z\text{ such that }tF+z\subset A\}$$ has nonempty interior. More generally, if $V$ is a topological vector space and $F=\{x_n\}_{n\in\mathbb{N}} \subset V$ is a bounded sequence, we show that if $A\subset V$ is non-meager and Baire, then $\Delta_F(A)$ has nonempty interior. The notion of boundedness in this context is described below. Note that the sequence $F$ can be countably infinite, which distinguishes this result from its measure-theoretic analogue. In the context of the topological version of Erd\H{o}s' similarity conjecture, we show that bounded countable sets are universal in non-meager Baire sets.

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arXiv2025

Counting Lattice Points near Kor\'anyi Spheres via Generalized Radon Transforms

In this note, we study a lattice point counting problem for spheres in Heisenberg groups, incorporating both the non-isotropic dilation structure and the non-commutative group law. More specifically, we establish an upper bound for the average number of lattice points in a $\delta$-neighborhood of a Kor\'anyi sphere of large radius, where the average considered is over Heisenberg group translations of the sphere. This is in contrast with previous works, which either count lattice points on dilates of a fixed sphere (see \cites{GNT, Gath2}) or consider averages over Euclidean translations of the sphere (see \cites{CT}). We observe that incorporating the Heisenberg group structure allows us to circumvent the degeneracy arising from the vanishing of the Gaussian curvature at the poles of the Kor\'anyi sphere. In fact, in lower dimensions (the first and second Heisenberg group), our method establishes an upper bound for this number which gives a logarithmic improvement over the bound implied by the previously known results. Even for the higher dimensional Heisenberg groups, we recover the bounds implied by the main result of \cite{GNT} using a completely different approach of generalized Radon transforms. Further, we obtain upper bounds for the average number of lattice points near more general spheres described with respect to radial, Heisenberg homogeneous norms as considered in \cite{GNT}.

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arXiv2024

Nonempty interior of configuration sets via microlocal partition optimization

We prove new results of Mattila-Sjölin type, giving lower bounds on Hausdorff dimensions of thin sets $E\subset \Bbb R^d$ ensuring that various $k$-point configuration sets, generated by elements of $E$, have nonempty interior. The dimensional thresholds in our previous work \cite{GIT20} were dictated by associating to a configuration function a family of generalized Radon transforms, and then optimizing $L^2$-Sobolev estimates for them over all nontrivial bipartite partitions of the $k$ points. In the current work, we extend this by allowing the optimization to be done locally over the configuration's incidence relation, or even microlocally over the conormal bundle of the incidence relation. We use this approach to prove Mattila-Sjölin type results for (i) areas of subtriangles determined by quadrilaterals and pentagons in a set $E\subset\Bbb R^2$; (ii) pairs of ratios of distances of 4-tuples in $\Bbb R^d$; and (iii) similarity classes of triangles in $\Bbb R^d$, as well as to (iv) give a short proof of Palsson and Romero Acosta's result on congruence classes of triangles in $\Bbb R^d$.

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arXiv2024

Realizing trees of configurations in thin sets

Let $\phi(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $\alpha>0$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $L^2({\mathbb R}^d) \to L^2_{\alpha}({\mathbb R}^d)$ for all $t>0$. Let $E$ be a compact subset of ${\mathbb R}^d$ for some $d \ge 2$, and suppose that the Hausdorff dimension of $E$ is $>d-\alpha$. We show that any tree graph $T$ on $k+1$ ($k \ge 1$) vertices is \new{stably} realizable in $E$, in the sense that \new{for each $t$ in some open interval} there exist distinct $x^1, x^2, \dots, x^{k+1} \in E$ %and $t>0$ such that the $\phi$-distance $\phi(x^i, x^j)=t$ for all pairs $(i,j)$ corresponding to the edges of $T$. We extend this result to trees whose edges are prescribed by more complicated point configurations, such as congruence classes of triangles.

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arXiv2023

Prescribed projections and efficient coverings by curves in the plane

Davies efficient covering theorem states that an arbitrary measurable set $W$ in the plane can be covered by full lines so that the measure of the union of the lines has the same measure as $W$. This result has an interesting dual formulation in the form of a prescribed projection theorem. In this paper, we formulate each of these results in a nonlinear setting and consider some applications. In particular, given a measurable set $W$ and a curve $\Gamma=\{(t,f(t)): t\in [a,b]\}$, where $f$ is $C^1$ with strictly monotone derivative, we show that $W$ can be covered by translations of $\Gamma$ in such a way that the union of the translated curves has the same measure as $W$. This is achieved by proving an equivalent prescribed generalized projection result, which relies on a Venetian blind construction.

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arXiv2022

Infinite constant gap length trees in products of thick Cantor sets

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to be countably infinite, which further distinguishes this work from previous results on patterns in fractal sets. This builds on the authors' previous work on graphs and distance sets of products of Cantor sets of sufficient Newhouse thickness.

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arXiv2022

Dimension and measure of sums of planar sets and curves

Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of $A+Γ:=\left\{a+v:a\in A, v\in Γ\right\}$ when $A\subset \mathbb{R}^2$ and $Γ$ is a piecewise $\mathcal{C}^2$ curve. Assuming $Γ$ has non-vanishing curvature, we verify that (a) if $\dim_{\rm H} A \leq 1$, then $\dim_{\rm H} (A+Γ)=\dim_{\rm H} A +1$, where $\dim_{\rm H}$ denotes the Hausdorff dimension; (b) if $\dim_{\rm H} A>1$, then $Leb_2(A+Γ)>0$, where $Leb_2$ denotes the $2$-dimensional Lebesgue measure; (c) if $\dim_{\rm H} A=1$ and $H^1(A) < \infty$, then $Leb_2(A+Γ)=0$ if and only if $A$ is an irregular (purely unrectifiable) $1$-set. Here, $H^1$ denotes the $1$-dimensional Hausdorff measure. Items (a) and (b) follow from previous works of Wolff and Oberlin using Fourier analysis. In this article, we develop an approach using nonlinear projection theory which gives new proofs of (a) and (b) and the first proof of (c). Item (c) has a number of consequences: if a circle is thrown randomly on the plane, it will almost surely not intersect the four corner Cantor set. Moreover, the pinned distance set of an irregular $1$-set has $1$-dimensional Lebesgue measure equal to zero at almost every pin $t\in \mathbb{R}^2$.

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arXiv2022

Lattice Points Close to the Heisenberg Spheres

We study a lattice point counting problem for spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms $\|(z,t)\|_α= ( |z|^α+ |t|^{α/2})^{1/α}$ for $α\geq 2$. As a first step, we reduce our counting problem to one of bounding an energy integral. The primary new challenges that arise are the presence of vanishing curvature and uneven dilations. In the process, we establish bounds on the Fourier transform of the surface measures arising from these norms. Further, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.

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arXiv2021

Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma

In this paper we prove that the set of tuples of edge lengths in $K_1\times K_2$ corresponding to a finite tree has non-empty interior, where $K_1,K_2\subset \mathbb{R}$ are Cantor sets of thickness $τ(K_1)\cdot τ(K_2) >1$. Our method relies on establishing that the pinned distance set is robust to small perturbations of the pin. In the process, we prove a nonlinear version of the classic Newhouse gap lemma, and show that if $K_1,K_2$ are as above and $ϕ: \mathbb{R}^2\times \mathbb{R}^2 \rightarrow \mathbb{R} $ is a function satisfying some mild assumptions on its derivatives, then there exists an open set $S$ so that $\bigcap_{x \in S} ϕ(x,K_1\times K_2)$ has non-empty interior.

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arXiv2021

Transversal families of nonlinear projections and generalizations of Favard length

Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a \textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.

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