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Idris Assani

Publications and source records attributed to Idris Assani.

At least 19 recordsLinked to original sources
arXiv2025

On Robin's Inequality and the Kaneko-Lagarias Inequality

We provide new, elementary proofs that Robin's inequality and the Lagarias inequality hold for almost every number, including all numbers not divisible by one of the prime numbers $2$, $3$, $5$; all primorials; given $k$ a natural number, all sufficiently large numbers of the form $2^kn$ for $n\ge1$ odd; and all $21$-free integers. Additionally, we prove that the Kaneko-Lagarias inequality holds for all natural numbers if and only if it holds for all superabundant numbers.

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arXiv2024

Higher Order Wiener-Wintner systems: examples and applications

We will construct ``higher-dimensional" versions of the Wiener-Wintner dynamical system that was originally studied by I. Assani in 2003. We will show that on these systems we can provide very simple proofs of the a.e. convergence of the multiple recurrence averages, as well as the multiple recurrence return times averages. We will do so by obtaining a quantitative control of the multiple ergodic averages by extending the estimate for the double recurrence that was attained by J. Bourgain. We will also observe that this class of dynamical systems contains numerous examples that are not bounded by the standard classifications (e.g. entropy, mixing), such as Kolmogorov systems, classical skew products, as well as systems for which the a.e. convergence of multiple recurrence is not currently known. Along our way, we will also provide alternative characteristics of the Host-Kra-Ziegler factors from the point of view of the uniform Wiener-Wintner theorem.

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arXiv2023

On the Convergence of the Density of Terras' Set

The Collatz Conjecture's connection to dynamical systems opens it to a variety of techniques aimed at recurrence and density results. First, we turn to density results and strengthen the result of Terras through finding a strict rate of convergence. This rate gives a preliminary result on the Triangle Conjecture, which describes a set nodes that would dominate $L^{C} = \{y \in \N \, | \, T^{k}(y) > y , \,\forall k \in \N \}$. Second, we extend prior arguments to show that the construction of several classes of measures imply the bounded trajectories piece of the Collatz Conjecture.

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arXiv2022

Syracuse Maps as Non-singular Power-Bounded Transformations and Their Inverse Maps

We prove that the dynamical system $(\mathbb{N}, 2^{\mathbb{N}}, T, \mu)$, where $\mu$ is a finite measure equivalent to the counting measure, is power-bounded in $L^1(\mu)$ if and only if there exists one cycle of the map $T$ and for any $x \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that $T^k(x)$ is in some cycle of the map $T$. This result has immediate implications for the Collatz Conjecture, and we use it to motivate the study of number theoretic properties of the inverse image $T^{-1}(x)$ for $x \in \mathbb{N}$, where $T$ denotes the Collatz map here. We study similar properties for the related Syracuse maps, comparing them to the Collatz map. We also analyze some structural properties of the inverse image in relation to asymptotic density of the set $\{x \in \mathbb{N} \mid \exists k \in \mathbb{N}: T^k(x) < x\}$.

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arXiv2022

Collatz map as a non-singular transformation

Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ by $T(n) = \frac{n}{2} $ if $n$ is even and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\N, 2^{\N}, T,\mu)$ where $\mu$ is the counting measure. This dynamical system $(\N, 2^{\N}, T, \mu)$ has the following properties. \begin{enumerate} \item There exists an invariant finite measure $\gamma$ such that $\gamma(A) \leq \mu(A) $ for all $A \subset \N.$ \item For each function $f\in L^1(\mu)$ the averages $\frac{1}{N} \sum_{n=1}^N f(T^nx)$ converge for every $x\in \N$ to $ f^*(x)$ where $ f^* \in L^1(\mu).$ \end{enumerate} We also show that the Collatz conjecture is equivalent to the existence of a finite measure $\nu$ on $(\N, 2^{\N})$ making the operator $Vf = f\circ T$ power bounded in $L^1(\nu)$ with conserrvative part $\{1,2\}.$

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arXiv2021

Spatial-Temporal Differentiation Theorems

Let $(X, \mathcal{B}, \mu, T)$ be a dynamical system where $X$ is a compact metric space with Borel $\sigma$-algebra $\mathcal{B}$, and $\mu$ is a probability measure that's ergodic with respect to the homeomorphism $T : X \to X$. We study the following differentiation problem: Given $f \in C(X)$ and $F_k \in \mathcal{B}$, where $\mu(F_k) > 0$ and $\mu(F_k) \to 0$, when can we say that $$\lim_{k \to \infty} \frac{\int_{F_k} \left( \frac{1}{k} \sum_{i = 0}^{k - 1} T^i f \right) \mathrm{d} \mu}{\mu(F_k)} = \int f \mathrm{d} \mu ? $$

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arXiv2018

Coboundaries of nonconventional ergodic averages

Let $(X,\mathcal{A}, μ)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded functions $f_1, f_2, ..., f_H$ to be a coboundary. This condition turns out to be also necessary when one seeks bounded coboundaries.

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arXiv2016

Pointwise double recurrence and nilsequences

Consider a system $(X, \mathcal{F}, μ, T)$, bounded functions $f_1, f_2 \in L^\infty(μ)$ and $a,b \in \ZZ.$ We show that there exists a set of full measure $X_{f_1, f_2}$ in $X$ such that for all $x \in X_{f_1, f_2}$ and for every nilsequence $b_n$ , the averages \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)b_n \] converge. We will show that this can be deduced from the classical Wiener-Wintner theorem for the double recurrence theorem. Together with the past work on this subject, we will show that several statements regarding the extension of the double recurrence theorem are equivalent.

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arXiv2015

Extension of Wiener-Wintner double recurrence theorem to polynomials

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)ϕ(p(n)) \] converge for any polynomial $p$ with real coefficients, and any continuous function $ϕ$ from the torus to the set of complex numbers . We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial $p$, then the averages converge to zero uniformly for all polynomials. This paper combines the authors' previously announced work.

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arXiv2015

A good universal weight for nonconventional ergodic averages in norm

We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system $(X, \mathcal{F}, μ, T)$ and bounded functions $f_1, f_2 \in L^\infty(μ)$, there exists a set of full-measure $X_{f_1, f_2}$ in $X$ that is independent of integers $a$ and $b$ and a positive integer $k$ such that for all $x \in X_{f_1, f_2}$ and for every other measure-preserving system $(Y, \mathcal{G}, ν, S)$, and each bounded and measurable function $g_1, \ldots, g_k \in L^\infty(ν)$, the averages \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)g_1 \circ S^n g_2 \circ S^{2n} \cdots g_k \circ S^{kn} \] converge in $L^2(ν)$.

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arXiv2015

Pointwise recurrence for commuting measure preserving transformations

Let $(X,\mathcal{A}, μ)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages $$\frac{1}{N}\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\cdots f_H(T_H^nx)$$ converge a.e. for every function $f_i \in L^{\infty}(μ)$ .\\ As a consequence if $T_i = T^i$ for $1\leq i \leq H$ where $T$ is an invertible measure preserving transformation on $(X, \mathcal{A}, μ)$ then the averages $$\frac{1}{N}\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ converge a.e. This solves a long open question on the pointwise convergence of nonconventional ergodic averages. For $H=2$ it provides another proof of J. Bourgain's a.e. double recurrence theorem.

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arXiv2015

A good universal weight for multiple recurrence averages with commuting transformations in norm

We will show that the sequences appearing in Bourgain's double recurrence result are good universal weights to the multiple recurrence averages with commuting measure-preserving transformations in norm. This will extend the pointwise converge result of Bourgain, the norm convergence result of Tao, and the authors' previous work on the single measure-preserving transformation.

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arXiv2014

Pointwise characteristic factors for Wiener Wintner double recurrence theorem

In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let $(X, \mathcal{F}, μ, T)$ be a standard ergodic system. We will show that for any $f_1, f_2 \in L^\infty(X)$, the double recurrence Wiener-Wintner average \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x) e^{2πi n t} \] converges off a single null set of $X$ independent of $t$ as $N \to \infty$. Furthermore, we will show a uniform Wiener-Wintner double recurrence result: If either $f_1$ or $f_2$ belongs to the orthogonal complement of the Conze-Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

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arXiv2012

A Survey of the Return Times Theorem

The goal of this paper is to survey the history, development and current status of the Return Times Theorem and its many extensions and variations. Let $(X, \mathcal{F}, μ)$ be a finite measure space and let $T:X \rightarrow X$ be a measure preserving transformation. Perhaps the oldest result in ergodic theory is that of Poincaré's Recurrence Principle which states: For any set $A \in \mathcal{F}$, the set of points $x$ of $A$ such that $T^nx$ is not in the set $A$ for all $n > 0$ has zero measure. This says that almost every point of $A$ returns to $A$. In fact, almost every point of $A$ returns to $A$ infinitely often. The return time for a given element $x \in A$, $r_A(x) = \inf\{k \geq 1: T^kx \in A\}$, is the first time that the element $x$ returns to the set $A$. By Poincaré's Recurrence Principle there is set of full measure in $A$ such that all elements of this set have a finite return time. Our study of the Return Times Theorem asks how we can further generalize this notion. The paper begins by looking at early work with the concept of weighted averages. The second portion of the paper focuses on the historical development of the proofs of the Return Times Theorem. Three areas of extension of the Return Times Theorem are then considered: a multiterm version, characteristic factors and breaking the Hölderian duality. The paper concludes with discussion of some more recent work and open questions to consider.

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arXiv2008

The $(L^{1},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages

Let $(X,\mathcal{B}, μ, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\dis R^*:(f, g) \in L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^{p},L^{q})$ functions with $\frac{1}{p}+\frac{1}{q}\leq 1.$

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