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Bernard Host

Publications and source records attributed to Bernard Host.

At least 19 recordsLinked to original sources
arXiv2024

Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$

A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$ is a real polynomial with at least one non-constant irrational coefficient, then the set $\{P(2^m3^n)\colon m,n\in \mathbb{N}\}$ is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.

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arXiv2021

Multiple recurrence and convergence without commutativity

We establish multiple recurrence and convergence results for pairs of zero entropy measure preserving transformations that do not satisfy any commutativity assumptions. Our results cover the case where the iterates of the two transformations are $n$ and $n^k$ respectively, where $k\geq 2$, and the case $k=1$ remains an open problem. Our starting point is based on the observation that Furstenberg systems of sequences of the form $(f(T^{n^k}x))$ have very special structural properties when $k\geq 2$. We use these properties and some disjointness arguments in order to get characteristic factors with nilpotent structure for the corresponding ergodic averages, and then finish the proof using some equidistribution results on nilmanifolds.

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arXiv2019

Furstenberg systems of bounded multiplicative functions and applications

We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the Möbius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Teräväinen related to the odd order cases of the Chowla conjecture.

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arXiv2019

The logarithmic Sarnak conjecture for ergodic weights

The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Möbius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.

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arXiv2019

Asymptotic pairs in positive-entropy systems

We show that in a topological dynamical system $(X,T)$ of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs $(x,y)$ such that $x\not= y$ and $\lim_{n\to +\infty} d(T^n x,T^n y)=0$. More precisely we consider a $T$-ergodic measure $μ$ of positive entropy and prove that the set of points that belong to a proper asymptotic pair is of measure $1$. When $T$ is invertible, the stable classes (i.e., the equivalence classes for the asymptotic equivalence) are not stable under $T^{-1}$: for $μ$-almost every $x$ there are uncountably many $y$ that are asymptotic with $x$ and such that $(x,y)$ is a Li-Yorke pair with respect to $T^{-1}$. We also show that asymptotic pairs are dense in the set of topological entropy pairs.

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arXiv2016

Weighted multiple ergodic averages and correlation sequences

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the averages of this sequence times any nilsequence converge. Key role in the proof play two decomposition results of independent interest. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and B. Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use the previous results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

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arXiv2016

Higher order Fourier analysis of multiplicative functions and applications

We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of an arbitrary degree. The proof uses tools from higher order Fourier analysis and some soft number theoretic input that comes in the form of an orthogonality criterion of Kátai. We use variants of this structure theorem to derive applications of number theoretic and combinatorial flavor: $(i)$ we give simple necessary and sufficient conditions for the Gowers norms (over $\mathbb{N}$) of a bounded multiplicative function to be zero, $(ii)$ generalizing a classical result of Daboussi and Delange we prove asymptotic orthogonality of multiplicative functions to "irrational" nilsequences, $(iii)$ we prove that for certain polynomials in two variables all "aperiodic" multiplicative functions satisfy Chowla's zero mean conjecture, $(iv)$ we give the first partition regularity results for homogeneous quadratic equations in three variables showing for example that on every partition of the integers into finitely many cells there exist distinct $x,y$ belonging to the same cell and $λ\in \mathbb{N}$ such that $16x^2+9y^2=λ^2$ and the same holds for the equation $x^2-xy+y^2=λ^2$.

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arXiv2016

Asymptotics for multilinear averages of multiplicative functions

A celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalizes a well known convergence result of Wirsing. The key ingredients are a recent structural result for bounded multiplicative functions proved by the authors and the mean value theorem of Halász.

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arXiv2015

Multiple ergodic theorems for arithmetic sets

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter to those integers that have an even number of distinct prime factors, or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analysed using a recent structural result for bounded multiplicative functions proved by the authors.

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arXiv2015

Ergodic averages of commuting transformations with distinct degree polynomial iterates

We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$, where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and $T_1,...,T_\ell$ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.

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arXiv2014

Variations on topological recurrence

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems.

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arXiv2014

Uniformity of multiplicative functions and partition regularity of some quadratic equations

Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove partition regularity for certain equations involving forms in three variables, showing for example that the equations $16x^2+9y^2=n^2$ and $x^2+y^2-xy=n^2$ are partition regular, where $n$ is allowed to vary freely in $\mathbb{N}$. For each such problem we establish a density analogue that can be formulated in ergodic terms as a recurrence property for actions by dilations on a probability space. Our key tool for establishing such recurrence properties is a decomposition result for multiplicative functions which is of independent interest. Roughly speaking, it states that the arbitrary multiplicative function of modulus 1 can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of degree three.

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arXiv2012

Complexity of Nilsystems and systems lacking nilfactors

Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise to this behavior. In the topological setting, nilsystems have a particular type of complexity of polynomial growth, where the polynomial (with explicit degree) is an asymptotic both from below and above. We also deduce several ergodic and topological applications of these results.

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arXiv2011

Extensions of Cantor minimal systems and dimension groups

Given a factor map $p : (X,T) \to (Y,S)$ of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups $K_0(X)/K_0(Y)$ in terms of intermediate extensions which are extensions of $(Y,S)$ by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of $(Y,S)$ embeds into a proper subgroup of the dimension group of $(X,T)$, yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups $H^n(X \mid Y)$ associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups $H^n(X \mid Y)$ are torsion groups. As a consequence we can now identify $H^0(X \mid Y)$ as the torsion group of $ K_0(X)/K_0(Y)$.

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arXiv2011

The polynomial multidimensional Szemerédi Theorem along shifted primes

If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for some prime $p$. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.

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arXiv2010

A point of view on Gowers uniformity norms

Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of focusing on the norms themselves, we study associated dual norms and dual functions. Combining this study with a variant of the Szemeredi Regularity Lemma, we give a decomposition theorem for dual functions, linking the dual norms to classical norms and indicating that the dual norm is easier to understand than the norm itself. Using the dual functions, we introduce higher order algebras that are analogs of the classical Fourier algebra, which in turn can be used to further characterize the dual functions.

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