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We consider translational integer tilings by finite sets $A\subset\mathbb{Z}$. We introduce a new method based on \emph{splitting}, together with a new combinatorial interpretation of some of the main tools from our earlier work. We also use splitting to prove the Coven-Meyerowitz conjecture for a new class of tilings $A\oplus B=\mathbb{Z}_M$. This includes tilings of period $M=p_1^{n_1}p_2^{n_2}p_3^{n_3}$ with $p_1>p_2^{n_2-1}p_3^{n_3-1}$, and tilings of period $M=p_1^{n_1}p_2^2p_3^2p_4^2$ with $p_1>p_2p_3p_4$, where $p_1,p_2,p_3,p_4$ are distinct primes and $n_1,n_2,n_3\in\mathbb{N}$. This is the second one of the two papers replacing version 1 of arXiv:2207.11809 (the first one is available as arXiv:2207.11809 v2). The main results of this paper (Theorem 1.2, Corollaries 1.4 and 1.5) and the intermediate results in Section 4.2 are all new and did not appear previously in arXiv:2207.11809 v1 or anywhere else. The material in Sections 3, 4.1, and 5 (splitting and the splitting formulation of the slab reduction) did appear in arXiv:2207.11809 v1 and has been removed from arXiv:2207.11809 v2. The results in Section 7 were included in arXiv:2207.11809 v1 and have been removed from arXiv:2207.11809 v2; the proofs are shorter and (we hope) more readable., Comment: 22 pages. Some of the results here appeared previously in arXiv:2207.11809 v1 and were removed from arXiv:2207.11809 v2. See abstract for details

If a finite set $A$ tiles the integers by translations, it also admits a tiling whose period $M$ has the same prime factors as $|A|$. We prove that the minimal period of such a tiling is bounded by $\exp(c(\log D)^2/\log\log D)$, where $D$ is the diameter of $A$. In the converse direction, given $\epsilon>0$, we construct tilings whose minimal period has the same prime factors as $|A|$ and is bounded from below by $D^{3/2-\epsilon}$. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture., Comment: 7 pages. Added a remark in Section 2 and corrected some typos

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., Comment: 20 pages

For $p$ prime, let $\mathcal{H}^n$ be the linear span of characteristic functions of hyperplanes in $(\mathbb{Z}/p^k\mathbb{Z})^n$. We establish new upper bounds on the dimension of $\mathcal{H}^n$ over $\mathbb{Z}/p\mathbb{Z}$, or equivalently, on the rank of point-hyperplane incidence matrices in $(\mathbb{Z}/p^k\mathbb{Z})^n$ over $\mathbb{Z}/p\mathbb{Z}$. Our proof is based on a variant of the polynomial method using binomial coefficients in $\mathbb{Z}/p^k\mathbb{Z}$ as generalized polynomials. We also establish additional necessary conditions for a function on $(\mathbb{Z}/p^k\mathbb{Z})^n$ to be an element of $\mathcal{H}^n$., Comment: 33 pages

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., Comment: 49 pages. This is the first one of the two papers replacing v1; see abstract for details. arXiv admin note: text overlap with arXiv:2106.14044

We improve a special case of the Lam-Leung lower bound on the number of elements in a vanishing sum of $N$-th roots of unity. Using this result, we extend the Favard length estimates due to Bond, {\L}aba, and Volberg to a new class of rational product Cantor sets in $\mathbb{R}^2$.

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., Comment: 50 pages. Minor corrections and updates. To appear in Forum of Mathematics - Pi

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., Comment: 77 pages. Various edits and corrections in response to the referee's report

For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha<\alpha'$ for which the associated maximal operator is bounded from $L^p_\delta (\mathbb{R})$ 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 $\alpha>0$, and have no Fourier decay. The proof is based on a decoupling inequality similar to that of Laba and Wang., Comment: 26 pages. Minor corrections, two references added

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 $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}{\pi} \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] 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 $p=2$., Comment: 29 pages, 8 figures. Minor revisions and updates

For any $\alpha\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\| \widehat{f d\mu} \|_{p} \leq C_p \| f \|_{L^2(\mu)}$ for all $p>2d/\alpha$. 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 $\alpha=d/k$ 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 $\Lambda(p)$ sets., Comment: 21 pages

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

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 $e_n = (0,\dots,0,1)$., Comment: 38 pages, small changes made in light of comments from the referees

We prove the range of exponents in the general $L^2$ Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak and Seeger is sharp for a large class of measures on $\mathbb{R}^d$. This extends to higher dimensions the sharpness result of Hambrook and {\L}aba.

Let $E \subseteq R^n$ be a closed set of Hausdorff dimension $\alpha$. 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, $\alpha$ 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 $R$., Comment: 46 pages

This is an expository paper detailing some of the recent advances on the problem, with emphasis on the number-theoretic method developed in my paper with Bond and Volberg for rational product sets (arXiv:1109.1031)., Comment: Submitted to the Proceedings of the 2012 Abel Symposium

We prove that the range of exponents in Mockenhaupt's restriction theorem for Salem sets, with the endpoint estimate due to Bak and Seeger, is optimal., Comment: The accepted version, to appear in GAFA. The final publication is available at http://link.springer.com/article/10.1007/s00039-013-0240-9

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, {\L}aba-Zhai, and Bond-Volberg., Comment: 42 pages. To appear in the American Journal of Mathematics. Copyright 2012 The Johns Hopkins University Press

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})., Comment: 15 pages; to appear in Combinatorics, Probability and Computing

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 $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the corresponding maximal operators are bounded on $L^p(\mathbb R)$ for $p > (1 + \epsilon)/(1 - \epsilon)$. 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., Comment: Revised version. The final version will appear in Duke Math. J

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., Comment: 16 pages

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$ contains non-trivial 3-term arithmetic progressions., Comment: 42 pages

We extend two well-known results in additive number theory, S\'ark\"ozy's theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restriction-type Fourier analytic argument of Green and Green-Tao., Comment: 22 pages

We prove Pach-Sharir type incidence theorems for a class of curves in R^n and surfaces in R^3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree., Comment: 16 pages

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean distance replaced by non-Euclidean norms in which the unit ball is a polygon with 2K sides. We prove that for any such norm there is a set of Hausdorff dimension K/(K-1) whose distance set has Lebesgue measure 0., Comment: 11 pages

We extend Wolff's "local smoothing" inequality to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat direction. An immediate consequence is the L^p boundedness of the corresponding Fourier multiplier operators., Comment: 35 pages; some background information corrected

Let X be a 2-dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX must be if X contains a well-distributed set whose distance set Delta satisfies the estimate |\Delta\cap[0,N]|

We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We verify several special cases of the relevant conjectures; in particular, we find necessary and sufficient conditions for a set A to tile the integers if A is a direct sum of cyclic subsets., Comment: 14 pages

A probability measure in R^d is called a spectral measure if it has an orthonormal basis consisting of exponentials. In this paper we study spectral Cantor measures. We establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral Cantor measure. This extends the earlier results of Jorgensen-Pedersen and Strichartz., Comment: 10 pages

We consider the problem of characterizing finite sets which tile the integers by translations. Coven and Meyerowitz (J. Algebra 1999) found necessary and sufficient conditions for a finite set A to tile the integers under the assumption that |A| has at most 2 distinct prime factors. The purpose of this article is to settle, for the first time, certain three-prime cases., Comment: 17 pages

We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic progressions of length 3., Comment: 8 pages

We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1, and give an example showing that there is no analogue of the tiling result in dimensions 3 and higher., Comment: 11 pages, 3 figures; added a counterexample in dimensions 3 and higher

We use geometrical combinatorics arguments, including the ``hairbrush'' and x-ray arguments of Wolff and the sticky/plany/grainy analysis of Katz, Laba, and Tao, to show that Besicovitch sets in R^n have Minkowski dimension at least (n+2)/2 + \eps_n for all n > 3, where \eps_n > 0 is an absolute constant depending only on n. This complements the results of Katz, Laba, and Tao, which established the same result for n=3, and of Bourgain and Katz-Tao, arithmetic combinatorics techniques to establish the result for n > 8. In contrast to previous work, our arguments will be purely geometric and do not require arithmetic combinatorics., Comment: 31 pages, 3 figures, submitted GAFA

We prove an x-ray estimate in general dimension which is stronger than the Kakeya estimates of Wolff. This generalizes an x-ray estimate in three dimensions which is also due to Wolff., Comment: 24 pages, no figures, submitted to Revista

A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + \epsilon for some absolute constant \epsilon > 0. One observation arising from the argument is that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which we call ``stickiness,'' ``planiness,'' and ``graininess.'' The purpose of this paper is to improve upon the known bounds for the Minkowski dimension of Besicovitch sets in three dimensions. As a by-product of the argument we obtain some strong conclusions on the structure of Besicovitch sets with almost-minimal Minkowski dimension., Comment: 64 pages, published version