Searcharxiv⌕ Search

arXiv subjects

Izabella Laba

Publications and source records attributed to Izabella Laba.

At least 19 recordsLinked to original sources
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.

math.CA↗
arXiv2024

The Coven-Meyerowitz tiling conditions for 3 prime factors: the even case

We consider finite sets $A\subset\mathbb{Z}$ tiles the integers by translations. By periodicity, any such tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. Building on por previous work, we prove that a tentative characterization of finite tiles proposed by Coven and Meyerowitz holds for all integer tilings of period $M=(p_ip_jp_k)^2$, where $p_i,p_j,p_k$ are distinct primes. This extends the main result of [15] (Invent. Math. 2023), where we assumed that $M$ is odd. We also improve parts of the argument from [15]. We have split the earlier (70-page) version into two papers. The current version (49 pages) is the first of the two. The main result is the same as in the previous version: we prove (T2) in the 3-prime even case. The second paper will be posted shortly as a new submission. It will have a new main result where we prove (T2) for a new class of tilings (proved very recently, not included in v1 of this paper). Splitting-related results from the earlier 70-page version of this paper have been moved there.

math.CO↗
arXiv2024

Keller properties for integer tiling

Keller's conjecture on cube tilings asserted that, in any tiling of $\mathbb{R}^d$ by unit cubes, there must exist two cubes that share a $(d-1)$-dimensional face. This is now known to be true in dimensions $d\leq 7$ and false for $d\geq 8$. In this article, we investigate analogues of Keller's conjecture for integer tilings.

math.CO↗
arXiv2022

The Coven-Meyerowitz tiling conditions for 3 odd prime factors

It is well known that if a finite set $A\subset\mathbb{Z}$ tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. We are interested in characterizing all finite sets $A\subset\mathbb{Z}$ that have this property. Coven and Meyerowitz (1998) proposed conditions (T1), (T2) that are sufficient for $A$ to tile, and necessary when the cardinality of $A$ has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of $|A|$. We prove that the Coven-Meyerowitz tiling condition (T2) holds for all integer tilings of period $M=(p_ip_jp_k)^2$, where $p_i,p_j,p_k$ are distinct odd primes. The proof also provides a classification of all such tilings.

math.CO↗
arXiv2022

Combinatorial and harmonic-analytic methods for integer tilings

A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some $M\in\mathbb{N}$ and $B\subset\mathbb{Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with $A$ and $B$. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids, and saturating spaces, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set $A$ containing certain configuration can tile a cyclic group $\mathbb{Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in a follow-up paper that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz.

math.CO↗
arXiv2018

On the maximal directional Hilbert transform

For any dimension $n \geq 2$, we consider the maximal directional Hilbert transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}π \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] The main result in this article asserts that for any exponent $p \in (1, \infty)$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \subseteq \mathbb S^{n-1}$, \[||\mathscr{H}_U||_{p \rightarrow p} \geq C_{p,n} \sqrt{\log \#U}, \] where $\#U$ denotes the cardinality of $U$. As a consequence, the maximal directional Hilbert transform associated with an infinite set of directions cannot be bounded on $L^p(\mathbb{R}^{n})$ for any $n\geq 2$ and any $p \in (1, \infty)$. This completes a result of Karagulyan, who proved a similar statement for $n=2$ and $p=2$.

math.CA↗
arXiv2018

Maximal operators and decoupling for $Λ(p)$ Cantor measures

For $2\leq p<\infty$, $α'>2/p$, and $δ>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $α<α'$ for which the associated maximal operator is bounded from $L^p_δ(\mathbb{R})$ to $L^p(\mathbb{R})$. Maximal theorems for fractal measures on the line were previously obtained by Laba and Pramanik. The result here is weaker in that we are not able to obtain $L^p$ estimates; on the other hand, our approach allows Cantor measures that are self-similar, have arbitrarily low dimension $α>0$, and have no Fourier decay. The proof is based on a decoupling inequality similar to that of Laba and Wang.

math.CA↗
arXiv2016

Decoupling and near-optimal restriction estimates for Cantor sets

For any $α\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $α$ such that the associated natural measure $μ$ obeys the restriction estimate $\| \widehat{f dμ} \|_{p} \leq C_p \| f \|_{L^2(μ)}$ for all $p>2d/α$. This range is optimal except for the endpoint. This extends the earlier work of Chen-Seeger and Shmerkin-Suomala, where a similar result was obtained by different methods for $α=d/k$ with $k\in\mathbb{N}$. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of $Λ(p)$ sets.

math.CA↗
arXiv2016

A Discrete Carleson Theorem Along the Primes with a Restricted Supremum

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_Λf(n) := \sup_{λ\in Λ} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2πi λp}}{p} |,\] where $\pm \mathbb{P} := \{ \pm p : p \text{ is a prime} \}$, and $Λ\subset [0,1]$. We give sufficient conditions on $Λ$, met by (finite unions of) lacunary sets, for this to be a bounded sublinear operator on $\ell^p(\mathbb{Z})$ for $\frac{3}{2} < p < 4$.

math.CA↗
arXiv2016

On polynomial configurations in fractal sets

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.

math.CA↗
arXiv2013

Finite configurations in sparse sets

Let $E \subseteq R^n$ be a closed set of Hausdorff dimension $α$. For $m \geq n$, let $\{B_1,\ldots,B_k\}$ be $n \times (m-n)$ matrices. We prove that if the system of matrices $B_j$ is non-degenerate in a suitable sense, $α$ is sufficiently close to $n$, and if $E$ supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of $m$ depending on $n$ and $k$, the set $E$ contains a translate of a non-trivial $k$-point configuration $\{B_1y,\ldots,B_ky\}$. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in $ R^n$ and isosceles right triangles in $R^2$). This can be viewed as a multidimensional analogue of an earlier result of Laba and Pramanik on 3-term arithmetic progressions in subsets of $R$.

math.CA↗
arXiv2013

Arithmetic progressions in sumsets and L^p-almost-periodicity

We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B contains an arithmetic progression of length at least about exp(c (alpha beta log N)^{1/2}). Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}).

math.NT↗
arXiv2012

Buffon's needle estimates for rational product Cantor sets

Let $S_\infty=A_\infty\times B_\infty$ be a self-similar product Cantor set in the complex plane, defined via $S_\infty=\bigcup_{j=1}^L T_j(S_\infty)$, where $T_j:\C\to\C$ have the form $T_j(z)=\frac1{L}z+z_j$ and $\{z_1,...,z_L\}=A+iB$ for some $A,B\subset\rr$ with $|A|,|B|>1$ and $|A||B|=L$. Let $S_N$ be the $L^{-N}$-neighbourhood of $S_\infty$, or equivalently (up to constants), its $N$-th Cantor iteration. We are interested in the asymptotic behaviour as $N\to\infty$ of the {\it Favard length} of $S_N$, defined as the average (with respect to direction) length of its 1-dimensional projections. If the sets $A$ and $B$ are rational and have cardinalities at most 6, then the Favard length of $S_N$ is bounded from above by $CN^{-p/\log\log N}$ for some $p>0$. The same result holds with no restrictions on the size of $A$ and $B$ under certain implicit conditions concerning the generating functions of these sets. This generalizes the earlier results of Nazarov-Perez-Volberg, Łaba-Zhai, and Bond-Volberg.

math.CA↗
arXiv2010

Maximal operators and differentiation theorems for sparse sets

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - ε$, $0 \leq ε< {1/3}$ for which the corresponding maximal operators are bounded on $L^p(\mathbb R)$ for $p > (1 + ε)/(1 - ε)$. As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis, in particular there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions. Updates: Andreas Seeger has provided an argument to the effect that our global maximal operators are in fact bounded on L^p(R) for all p>1; in particular, it follows that our differentiation theorems are also valid for all p>1. Furthermore, David Preiss has proved that no such differentiation theorems (let alone maximal estimates) can hold with p=1. These arguments are included in the new version. We have also improved the exposition in a number of places.

math.CA↗
arXiv2009

The Favard length of product Cantor sets

Nazarov, Peres and Volberg proved recently that the Favard length of the $n$-th iteration of the four-corner Cantor set is bounded from above by $n^{-c}$ for an appropriate $c$. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.

math.CA↗