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We study higher uniformity properties of the von Mangoldt function $\Lambda$, the M\"obius function $\mu$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $\Lambda^\sharp$ and $d_k^\sharp$ be suitable approximants of $\Lambda$ and $d_k$, $G/\Gamma$ a filtered nilmanifold, and $F\colon G/\Gamma \to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (\Lambda(n)-\Lambda^\sharp(n)) \overline{F}(g(n)\Gamma) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)\Gamma) \right| = o(H \log^{k-1} X). \] As a consequence, we show that the short interval Gowers norms $\|\Lambda-\Lambda^\sharp\|_{U^s(X,X+H]}$ and $\|d_k-d_k^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $II$ estimates obtained by developing a "contagion lemma" for nilsequences and then using this to "scale up" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity., Comment: 104 pages

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $\Lambda$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the M\"obius weight $\mu$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cram\'er'' and ''Heath-Brown'' type., Comment: 37 pages

We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erd\H{o}s. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns)., Comment: 7 pages, no figures

For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A: n, m \leq x} \frac{1}{\operatorname{lcm}(n,m)} \ll_{C_0} \left(\sum_{n \in A: n \leq x} \frac{1}{n}\right)^2$$ as $x \to \infty$, with the growth rate given here optimal up to the dependence on $C_0$. This answers in the negative a question of Erd\H{o}s and Graham, and also clarifies the nature of certain ``mostly coprime'' sets studied by Bergelson and Richter., Comment: 15 pages, no figures. Implements suggestions from referee, including simplified versions of main results

Fix $k \geq 2$. For any $N \geq 1$, let $F_k(N)$ denote the cardinality of the largest subset of $\{1,\dots,N\}$ that does not contain $k$ distinct elements whose product is a square. Erd\H{o}s, S\'ark\H{o}zy, and S\'os showed that $F_2(N) = (\frac{6}{\pi^2}+o(1)) N$, $F_3(N) = (1-o(1))N$, $F_k(N) \asymp N/\log N$ for even $k \geq 4$, and $F_k(N) \asymp N$ for odd $k \geq 5$. Erd\H{o}s then asked whether $F_k(N) = (1-o(1)) N$ for odd $k \geq 5$. Using a probabilistic argument, we answer this question in the negative., Comment: 13 pages, one figure. This is the final version, incorporating referee comments (and slightly more numerical data)

We prove a Freiman--Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let $G$ be an abelian group of torsion $m$ (meaning $mg=0$ for all $g \in G$) and suppose that $A$ is a non-empty subset of $G$ with $|A+A| \leq K|A|$. Then $A$ can be covered by at most $(2K)^{O(m^3)}$ translates of a subgroup of $H \leq G$ of cardinality at most $|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$ in a recent paper of the authors., Comment: 33 pages, to be submitted; v2 corrects some minor issues pointed out by readers of v1

We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic $2$. The argument extends to odd characteristic, with details to follow in a subsequent paper., Comment: 33 pages, updated version now to be submitted for publication

The classical Maclaurin inequality asserts that the elementary symmetric means $$ s_k(y) = \frac{1}{\binom{n}{k}} \sum_{1 \leq i_1 < \dots < i_k \leq n} y_{i_1} \dots y_{i_k}$$ obey the inequality $s_\ell(y)^{1/\ell} \leq s_k(y)^{1/k}$ whenever $1 \leq k \leq \ell \leq n$ and $y = (y_1,\dots,y_n)$ consists of non-negative reals. We establish a variant $$ |s_\ell(y)|^{\frac{1}{\ell}} \ll \frac{\ell^{1/2}}{k^{1/2}} \max (|s_k(y)|^{\frac{1}{k}}, |s_{k+1}(y)|^{\frac{1}{k+1}})$$ of this inequality in which the $y_i$ are permitted to be negative. In this regime the inequality is sharp up to constants. Such an inequality was previously known without the $k^{1/2}$ factor in the denominator., Comment: 12 pages, no figures. Minor typo corrections

In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in $\mathbb{Z}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually $\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2\times G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in $\mathbb{Z}^d$, $d\geq 3$., Comment: 44 pages, 10 figures, typos corrected

Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $\varphi(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) \pi(x)$ for all $x \geq 10$, answering questions of Erd\H{o}s and Pollack--Pomerance--Trevi\~no. A similar result is also obtained for the sum of divisors function $\sigma(n)$., Comment: 23 pages. This is the final version, addressing referee comments

We give an improved lower bound for the average of the Erd\H{o}s-Hooley function $\Delta(n)$, namely $\sum_{n\le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $\eta = 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of $x (\log\log x)^{11/4}$ of the second and third authors., Comment: 15 pages. Added remarks in the end of page 3; added references [3] and [9]. Final version, to appear in Proc. London Math. Soc

It is an open question of Erd\H{o}s as to whether the alternating series $\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$ is (conditionally) convergent, where $p_n$ denotes the $n^{\mathrm{th}}$ prime. By using a random sifted model of the primes recently introduced by Banks, Ford, and the author, as well as variants of a well known calculation of Gallagher, we show that the answer to this question is affirmative assuming a suitably strong version of the Hardy--Littlewood prime tuples conjecture., Comment: 16 pages, no figures. A generalization to the main result added, as well as some more references and open questions

The Brascamp-Lieb inequalities are a generalization of the H\"older, Loomis-Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper we introduce an "adjoint" version of these inequalities, which can be viewed as an $L^p$ version of the entropy Brascamp-Lieb inequalities of Carlen and Cordero-Erausquin. As applications, we reprove a log-convexity property of the Gowers uniformity norms, and establish some reverse $L^p$ inequalities for various tomographic transforms. We conclude with some open questions., Comment: 43 pages; some further references and remarks added

The entropic doubling $\sigma_{\operatorname{ent}}[X]$ of a random variable $X$ taking values in an abelian group $G$ is a variant of the notion of the doubling constant $\sigma[A]$ of a finite subset $A$ of $G$, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of P\'alv\"olgyi and Zhelezov on the ``skew dimension'' of subsets of $\mathbf{Z}^D$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of $\mathbf{Z}^D$ with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$ implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$., Comment: 37 pages

Associated to two given sequences of eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$ and $\mu_1 \geq \dots \geq \mu_n$ is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as $n \rightarrow \infty$. Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand--Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni--Erd\H{o}s--Schr\"oder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process., Comment: 45 pages, 15 figures

The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u

Let $\Gamma$ be a countable abelian group, let $k\geq 1$, and let $\mathrm{X}=(X,\mathcal{X},\mu,T)$ be an ergodic $\Gamma$-system of order $k$ in the sense of Host--Kra--Ziegler. The $\Gamma$-system $\mathrm{X}$ is said to be totally disconnected if all its structure groups are totally disconnected. We show that any totally disconnected $\Gamma$-system of order $k$ is a generalized factor of a $\mathbb{Z}^\omega$-system with the structure of a Weyl system. As a consequence of this structure theorem, we show that totally disconnected $\Gamma$-systems of order $k$ are represented by translations on double cosets of nilpotent Polish groups. By a correspondence principle of two of us, we can use this representation to establish a (weak) inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion., Comment: 96 pages

It was conjectured by Bergelson, Tao, and Ziegler that every Host--Kra ${\mathbf F}_p^\omega$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k \leq p+1$. In this paper we show that the conjecture fails when $k=5, p=2$. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: {\mathbf F}_2^n \to {\mathbf C}$ of large Gowers norm $\|f\|_{U^6({\mathbf F}_2^n)}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial $e(P)$, but with the property that all such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of $f$., Comment: 70 pages, no figures. Minor changes in notation (particularly with regards to the concept of a k-cocycle) and some new remarks

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$., Comment: 13 pages, no figures. A reference has been corrected

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic., Comment: 50 pages, 13 figures. Final version

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R^d}$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z^2} \times G_0$ for some finite abelian $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.

We study higher uniformity properties of the M\"obius function $\mu$, the von Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$. More precisely, letting $\Lambda^\sharp$ and $d_k^\sharp$ be suitable approximants of $\Lambda$ and $d_k$ and $\mu^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)\Gamma)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \] when $\theta = 5/8$ and $f \in \{\Lambda, \mu, d_k\}$ or $\theta = 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,\theta$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums., Comment: 103 pages; Some typo fixes and a slight fix in proof of Proposition 2.14 compared to the published version, acknowledgment added

Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma" that asserts, roughly speaking, that $F \odot A = X$ implies $F^r \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus $\mathbb{T}^d$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case)., Comment: 3 figures

A well known open problem of Meir and Moser asks if the squares of sidelength $1/n$ for $n \geq 2$ can be packed perfectly into a square of area $\sum_{n=2}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}-1$. In this paper we show that for any $1/2 < t < 1$, and any $n_0$ that is sufficiently large depending on $t$, the squares of sidelength $n^{-t}$ for $n \geq n_0$ can be packed perfectly into a square of area $\sum_{n=n_0}^\infty \frac{1}{n^{2t}}$. This was previously known (if one packs a rectangle instead of a square) for $1/2 < t \leq 2/3$ (in which case one can take $n_0=1$)., Comment: 11 pages, 1 figure. Several minor corrections

We state and prove a quantitative inverse theorem for the Gowers uniformity norm $U^3(G)$ on an arbitrary finite abelian group $G$; the cases when $G$ was of odd order or a vector space over ${\mathbf F}_2$ had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host--Kra type for ergodic ${\mathbf Z}^\omega$-actions of order $2$ on probability spaces established recently by Shalom and the authors., Comment: 48 pages, no figures. This is the published version

Let $\Gamma$ be a countable abelian group. An (abstract) $\Gamma$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\Gamma$ - is said to be a Conze-Lesigne system if it is equal to its second Host-Kra-Ziegler factor $\mathrm{Z}^2(\mathrm{X})$. The main result of this paper is a structural description of such Conze-Lesigne systems for arbitrary countable abelian $\Gamma$, namely that they are the inverse limit of translational systems $G_n/\Lambda_n$ arising from locally compact nilpotent groups $G_n$ of nilpotency class $2$, quotiented by a lattice $\Lambda_n$. Results of this type were previously known when $\Gamma$ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers $U^3(G)$ norm for arbitrary finite abelian groups $G$., Comment: 69 pages, [v2]: Final version accepted for publication in Communications of the AMS

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}\Lambda(n+h_1)\cdots \Lambda(n+h_k)\lambda(n+h_{1}')\cdots \lambda(n+h_{\ell}')$$ for any $0\leq k\leq 2$ and $\ell \geq 0$. Specializing to either $\ell=0$ or $k=0$, we deduce the previously known results on the Hardy--Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results., Comment: 54 pages, no figures. To appear in J. London Math. Soc

We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can be tiled by translations of $F_1,F_2$. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of $E$ by the tiles $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $\mathbb{Z}^2$). A similar construction also applies for $G = \mathbb{Z}^d$ for sufficiently large $d$. If one allows the group $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile $F$. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles., Comment: Revised version incorporating referee's suggestions, minor typos corrected

We determine the optimal inequality of the form $\sum_{k=1}^m a_k\sin kx\leq 1$, in the sense that $\sum_{k=1}^m a_k$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle., Comment: 19 pages, 6 figures; v2: Omitted the application to algebraic varieties of general type with small volume, in view of better results for that problem in arXiv:2109.13383

We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\"obius function $\mu$ and the von Mangoldt function $\Lambda$ for all $k$, with error terms of shape $O((\log\log N)^{-c})$. As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no $k$-term arithmetic progressions with shifted prime difference., Comment: 57 pages; Further referee comments incorporated; to appear in J. Eur. Math. Soc

Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon > 0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation $(n)_m = t$, where $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling factorial., Comment: 33 pages

In the presence of a confining potential $V$, the eigenfunctions of a continuous Schr\"odinger operator $-\Delta +V$ decay exponentially with the rate governed by the part of $V$ which is above the corresponding eigenvalue; this can be quantified by a method of Agmon. Analogous localization properties can also be established for the eigenvectors of a discrete Schr\"odinger matrix. This note shows, perhaps surprisingly, that one can replace a discrete Schr\"odinger matrix by \emph{any} real symmetric $Z$-matrix and still obtain eigenvector localization estimates. In the case of a real symmetric non-singular $M$-matrix $A$ (which is a situation that arises in several contexts, including random matrix theory and statistical physics), the \emph{landscape function} $u = A^{-1} 1$ plays the role of an effective potential of localization. Starting from this potential, one can create an Agmon-type distance function governing the exponential decay of the eigenfunctions away from the "wells" of the potential, a typical eigenfunction being localized to a single such well.

Sendov's conjecture asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all of its zeroes in closed unit disk $\{ z: |z| \leq 1 \}$, then for each such zero $\lambda_0$ there is a zero of the derivative $f'$ in the closed unit disk $\{ z: |z-\lambda_0| \leq 1 \}$. This conjecture is known for $n < 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov's conjecture holds for $n \geq n_0$. For $\lambda_0$ away from the origin and the unit circle we can appeal to the prior work of D\'egot and Chalebgwa; for $\lambda_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $\lambda_0$ is extremely close to the unit circle); and for $\lambda_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle., Comment: 38 pages, 5 figures. To appear, Acta Mathematica. Referee comments incorporated

We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions., Comment: 38 pages, 5 figures. This is the publication version, incorporating the journal style

We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.

We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools are useful in applications to "uncountable" ergodic theory (as demonstrated by the authors and others)., Comment: 102 pages, 33 diagrams, 1 table; [v3]: revision based on a thorough referee report, several notational changes

The Mackey-Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by a compact group $K$, by showing that such extensions are isomorphic to a group skew-product $X \equiv Y \rtimes_\rho H$ for some closed subgroup $H$ of $K$. An analogous theorem is also available for ergodic homogeneous extensions $X$ of $Y$, namely that they are isomorphic to a homogeneous skew-product $Y \rtimes_\rho H/M$. These theorems have many uses in ergodic theory, for instance playing a key role in the Host-Kra structural theory of characteristic factors of measure-preserving systems. The existing proofs of the Mackey-Zimmer theorem require various "countability", "separability", or "metrizability" hypotheses on the group $\Gamma$ that acts on the system, the base space $Y$, and the group $K$ used to perform the extension. In this paper we generalize the Mackey-Zimmer theorem to "uncountable" settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a "canonical model" for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host-Kra structural theory., Comment: 56 pages, 8 figures; [v4]: final version accepted for publication in Studia Math. Also modified the notation in line with referee comments on this paper and related papers