Hardy Field Research Articles

Density modulo 1 and Hardy fields

AbstractIn this article we extend Boshernitzan’s result on density modulo 1 of sequences arising from functions belonging to a Hardy field. We also merge these results with Furstenberg’s ×2×3 theorem. We prove, for example, that, given a vector f of subpolynomial functions in a Hardy field, such that $$(\mathbf{f}(n))_{n=1}^{\infty}$$ ( f ( n ) ) n = 1 ∞ is dense modulo 1 in Rd, the sequence (2m3nα, f(n))m,n≥1 is dense modulo 1 in Rd+1 for irrational α. Some negative results concerning Furstenberg’s theorem are obtained as well.

On finite pseudorandom binary sequences: functions from a Hardy field

On finite pseudorandom binary sequences: functions from a Hardy field

Pointwise convergence of multiple ergodic averages – PCMEA

Pointwise convergence of multiple ergodic averages – PCMEA Given an arbitrary measure preserving system we show that the multilinear ergodic averages sampled along an arbitrary number of sequences coming from a Hardy field converge pointwise almost everywhere. We aim to prove this for as wide a class of Hardy field functions as possible. To do so, we establish a long variational inequality along lacunary sequences which implies a maximal inequality, norm convergence, and pointwise convergence. By a transference argument it suffices prove this long variational inequality in the case that the measure preserving system is the integers. This reduction allows us to use tools from discrete harmonic analysis, additive combinatorics, and analytic number theory. We then give applications in areas such as upcrossings, equidistribution, and combinatorics.

Bracket words along Hardy field sequences

AbstractWe study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $ . We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.

Joint ergodicity of Hardy field sequences

We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions t 3 / 2 , t log ⁡ t t^{3/2}, t\log t and e log ⁡ t e^{\sqrt {\log t}} . We show that if all non-trivial linear combinations of the functions a 1 a_1 , …, a k a_k stay logarithmically away from rational polynomials, then the L 2 L^2 -limit of the ergodic averages 1 N ∑ n = 1 N f 1 ( T ⌊ a 1 ( n ) ⌋ x ) ⋅ ⋯ ⋅ f k ( T ⌊ a k ( n ) ⌋ x ) \frac {1}{N} \sum _{n=1}^{N}f_1(T^{\lfloor {a_1(n)}\rfloor }x)\cdot \dots \cdot f_k(T^{\lfloor {a_k(n)}\rfloor }x) exists and is equal to the product of the integrals of the functions f 1 f_1 , …, f k f_k in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions a 1 a_1 , …, a k a_k , we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.

Uniform distribution in nilmanifolds along functions from a Hardy field

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if X = G/Γ is a nilmanifold, a1, …, ak ∈ G are commuting nilrotations, and f1, …, fk are functions of polynomial growth from a Hardy field then we show that

Furstenberg systems of Hardy field sequences and applications

We study measure preserving systems, called Furstenberg systems, that model the statistical behavior of sequences defined by smooth functions with at most polynomial growth. Typical examples are the sequences (\({n^{{3 \over 2}}}\)), (n log n), and (\([{n^{{3 \over 2}}}]\)), α ∈ ℝ ℚ, where the entries are taken mod 1. We show that their Furstenberg systems arise from unipotent transformations on finite-dimensional tori with some invariant measure that is absolutely continuous with respect to the Haar measure and deduce that they are disjoint from every ergodic system. We also study similar problems for sequences of the form (\(g({S^{[{n^{{3 \over 2}}}]}}y)\)), where S is a measure preserving transformation on the probability space (Y, ν), g ∈ L∞(ν), and y is a typical point in Y. We prove that the corresponding Furstenberg systems are strongly stationary and deduce from this a multiple ergodic theorem and a multiple recurrence result for measure preserving transformations of zero entropy that do not satisfy any commutativity conditions.

Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy

We introduce a notion of regular separation for solutions of systems of ODEs y'=F(x,y) , where F is definable in a polynomially bounded o-minimal structure and y=(y_1,y_2) . Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author.

Multiple ergodic averages for tempered functions

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the \begin{document}$ L^2 $\end{document} -norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the expected one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a nice class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case).

Single and multiple recurrence along non-polynomial sequences

We establish new recurrence and multiple recurrence results for a rather large family F of non-polynomial functions which contains tempered functions and (non-polynomial) functions from a Hardy field with polynomial growth.In particular, we show that, somewhat surprisingly (and in the contrast to the multiple recurrence along polynomials), the sets of return times along functions from F are thick, i.e., contain arbitrarily long intervals. A major component of our paper is a new result about equidistribution of sparse sequences on nilmanifolds, whose proof borrows ideas from the work of Green and Tao [26].Among other things, we show that for any f∈F, any invertible probability measure preserving system (X,B,μ,T), any A∈B with μ(A)>0, and any ε>0, the sets of returns{n∈N:μ(A∩T−⌊f(n)⌋A)>μ2(A)−ε}{n∈N:μ(A∩T−⌊f(n)⌋A∩T−⌊f(n+1)⌋A∩⋯∩T−⌊f(n+k)⌋A)>0} are thick.Our recurrence theorems imply, via Furstenberg's correspondence principle, some new combinatorial results. For example, we show that given a set E⊂N with positive upper density, for every k∈N there are a,n∈N such that{a,a+⌊f(n)⌋,⋯,a+⌊f(n+k)⌋}⊂E. When f(n)=nc, with c>0 non-integer, this result provides a positive answer to a question posed by Frantzikinakis [16, Problem 23].

Ilyashenko algebras based on transserial asymptotic expansions

We construct a Hardy field that contains Ilyashenko's class of germs at +∞ of almost regular functions found in [12] as well as all log-exp-analytic germs. This implies non-oscillatory behaviour of almost regular germs with respect to all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. [7]. As these series generally have support of order type larger than ω, the notion of asymptotic expansion itself needs to be generalized.

Optimal lower bounds for multiple recurrence

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form $$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$ for various families $\{f_{1},\ldots ,f_{k}\}$ of sequences $f_{i}:\mathbb{N}\rightarrow \mathbb{N}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_{n})$, where $q\in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1)=0$ and $p_{n}$ denotes the $n$th prime; or when $f$ is a certain Hardy field sequence. If $T^{q}$ is ergodic for some $q\in \mathbb{N}$, then, for all $r\in \mathbb{Z}$, $S$ is syndetic if $f(n)=qn+r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and, for $k=3$, we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

Uniform distribution of subpolynomial functions along primes and applications

Let H be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let f ∈ H be a subpolynomial function. Let P be the sequence of naturally ordered primes. We show that (f(n))n∈ℕ is uniformly distributed mod1 if and only if (f (p))p∈P is uniformly distributed mod 1. This result is then utilized to derive various ergodic and combinatorial statements which significantly generalize the results obtained in [BKMST].

The surreal numbers as a universal $H$-field

We show that the natural embedding of the differential field of transseries into Conway’s field of surreal numbers with the Berarducci–Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field of surreals with the Berarducci–Mantova derivation.

On convergence of oscillatory ergodic Hilbert transforms

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let $p(t)$ be a Hardy field function which grows super-linearly and stays sufficiently far from polynomials. We show that for each measure-preserving system, $(X,\Sigma,\mu,\tau)$, with $\tau$ a measure-preserving $\mathbb{Z}$-action, the modulated one-sided ergodic Hilbert transform \[ \sum_{n=1}^\infty \frac{e^{2\pi i p(n)}}{n} \tau^n f(x) \] converges $\mu$-a.e. for each $f \in L^r(X), 1 \leq r < \infty$. This affirmatively answers a question of J. Rosenblatt. In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert transforms, \[ \sum_{n=1}^\infty \frac{X_n}{n} \tau^n f(x), \] where $\{ X_n \}$ are uniformly bounded, independent, and mean-zero random variables.

Integer part independent polynomial averages and applications along primes

Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman (Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems, 25 (2005) no. 1, 201-213) and Frantzikinakis (Multiple recurrence and convergence for Hardy field sequences of polynomial growth. Journal d'Analyse Mathematique, 112 (2010), 79-135 and Equidistribution of sparse sequences on nilmanifolds. Journal d'Analyse Mathematique, 109 (2009), 353-395) we show that the ergodic averages with iterates given by the integer part of real-valued strongly independent polynomials, converge in the mean to the right-expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications while combining these results with methods from The polynomial multidimensional Szemeredi theorem along shifted primes. Israel J. Math., 194 (2013), no. 1, 331-348 and Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory Dynam. Systems, 1-20. doi:10.1017/etds.2016.40 we get the respective right limits and combinatorial results for multiple averages for a single sequence as well as for several sequences along prime numbers.

Pointwise multiple averages for sublinear functions

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

Quasianalytic Ilyashenko Algebras

AbstractWe construct a quasianalytic field of germs at +∞ of real functions with logarithmic generalized power series as asymptotic expansions, such that is closed under differentiation and log-composition; in particular, is a Hardy field. Moreover, the field o (−log) of germs at 0+ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

(Uniform) convergence of twisted ergodic averages

Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p&gt;1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

Bee colony optimization for the satisfiability problem in probabilistic logic

This paper presents the first heuristic method for solving the satisfiability problem in the logic with approximate conditional probabilities. This logic is very suitable for representing and reasoning with uncertain knowledge and for modeling default reasoning. The solution space consists of variables, which are arrays of 0 and 1 and the associated probabilities. These probabilities belong to a recursive non-Archimedean Hardy field which contains all rational functions of a fixed positive infinitesimal. Our method is based on the bee colony optimizationmeta-heuristic. The proposed procedure chooses variables from the solution space and determines their probabilities combining some other fast heuristics for solving the obtained linear system of inequalities. Experimental evaluation shows a high percentage of success in proving the satisfiability of randomly generated formulas. We have also showed great advantage in using a heuristic approach compared to standard linear solver.