arXiv subjects

Sohail Farhangi

Publications and source records attributed to Sohail Farhangi.

13 recordsLinked to original sources

arXiv2026

Solutions to 6 problems in recurrence and Van der Corput sets

We present solutions to six problems concerning sets of nice recurrence and van der Corput (vdC) sets. We answer a question of Moreira, repeated by Fish and Skinner, by showing that every set of ergodic nice recurrence is a set of nice recurrence. We answer questions of Bergelson as well as Bergelson and Lesigne by showing that every set of nice recurrence is a nice vdC-set, that the family of nice vdC-sets is partition regular, and that the family of sets of nice recurrence is partition regular. We mention that Chapman independently proved that sets of nice recurrence are partition regular. We answer a question of Kelly and Lê, a special case of which was previously asked by Peres, and show that for every nontrivial compact metrizable group $K$, every $K$-vdC set is also a vdC set. Lastly, we answer a question of Farhangi, Rodríguez, and Tucker-Drob and show that for any countably infinite discrete group $G$, a vdC set in $G$ is a set of operatorial recurrence in $G$. The initial solutions to these problems were produced by an AI system. The authors checked their correctness, simplified and reformulated them as clear, human-readable proofs, and situated the resulting contributions within the existing literature.

math.DS↗

arXiv2026

On the Jacobs-de Leeuw-Glicksberg decomposition for representations of semigroups

We develop a systematic study of the Jacobs-de Leeuw-Glicksberg decomposition for JdLG-admissible semigroup representations on Banach spaces, with emphasis on its structural, ergodic, and combinatorial aspects. For a JdLG-admissible representation $π$ of a semigroup $S$ on a Banach space $E$, we show that the reversible part is weakly equivalent to a unitary representation on a Hilbert space that decomposes as a direct sum of finite-dimensional representations. We also characterize the almost weakly stable part in terms of the unique invariant mean on the space of weakly almost periodic functions. When $S$ is a bi-amenable measured semigroup, we obtain further characterizations of the almost weakly stable part using invariant means and averages along Følner sequences. In addition, we describe, in terms of ultrafilters, the unique projection onto the reversible part whose kernel is the almost weakly stable part, and derive several combinatorial consequences. A range of examples are given to demonstrate the sharpness of our results. Many of our theorems extend known features of the compact-weak mixing decomposition for unitary operators; although some auxiliary results were previously known or part of the folklore, their extension to this general setting, particularly for nonseparable Banach spaces, requires new ideas.

math.FA↗

arXiv2026

Multipliers and Disjointness from Mixing

In 2005, Parreau proved that if a measure preserving system is not strongly mixing then it contains a non-trivial factor that is disjoint from every strongly mixing system. Taking this construction as the starting point, we develop the complementary notions of $\mathcal U$-generated and $\mathcal U$-mixing systems, for a set $\mathcal U$ of ultrafilters, and use them to recover several classical results in ergodic theory as special cases of a unified framework. We prove that a system is $\mathcal U$-mixing if and only if it is disjoint from all $\mathcal U$-generated systems. In fact, we show that if $\mathcal Y$ is a $\mathcal U$-generated system and $\mathcal Z$ is disjoint from every $\mathcal U$-mixing system, then any joining of $\mathcal Y$ and $\mathcal Z$ remains disjoint from all $\mathcal U$-mixing systems. We also show that every partially rigid system is a finite extension of some $\mathcal{U}$-generated system.

math.DS↗

arXiv2025

Asymptotic dynamics on amenable groups and van der Corput sets

We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the Følner sequence used to define it. This result has been discovered independently by Saúl Rodríguez Martín. Both ours and Rodríguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve Følner sequences remain equivalent for arbitrary countably infinite groups.

math.DS↗

arXiv2025

The Theory of Normality for Dynamically Generated Cantor Series Expansions

The theory of normality for base $g$ expansions of real numbers in $[0,1)$ is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion, $β$-expansions, and Lüroth series expansions. Let $Q=(q_n)_{n \in \mathbb{N}}$ be a sequence of integers greater than or equal to 2. The $Q$-Cantor series expansion of $x \in [0,1)$ is the unique sum of the form $x=\sum_{n=1}^\infty \frac{x_n}{q_1q_2\cdots q_n}$, where $x_n \neq q_n-1$ infinitely often. For the Cantor series expansions, most of the literature thus far considers $Q$ where the theory of normality differs drastically from that of the base $g$ expansions. We introduce the class of dynamically generated Cantor series expansions, which is a large class of Cantor series expansions for which much of the classical theory of base $g$ expansions can be developed in parallel. This class includes many examples such as the Thue-Morse sequence on $\{2,3\}$ and translated Champernowne numbers. A special case of our main results is that if $Q$ is a bounded basic sequence that is dynamically generated by an ergodic system having zero entropy, then normality base $Q$ coincides with distribution normality base $Q$, and $Q$ possesses a Hot Spot Theorem.

math.DS↗

arXiv2025

Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p \in \mathbb{Z}[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{Z}\setminus\{0\}$ is undecidable conditional on Hilbert's tenth problem for $\mathbb{Q}$. For other integral domains, we get the analogous result unconditionally. More generally, we determine the exact lightface complexity of the various sets of interest. For example, we show that the set of homogeneous polynomials $p \in \mathbb{F}_q(t)[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{F}_q(t)\setminus\{0\}$ is $Π_2^0$-complete. We also prove several other results of independent interest. These include a compactness principle and a uniformity principle for density Ramsey theory on countable cancellative left amenable semigroups, as well as the existence of the natural extension for measure preserving systems of countable cancellative left reversible semigroups.

math.LO↗

arXiv2025

Uniform vector-valued pointwise ergodic theorems for operators

We prove a uniform vector-valued Wiener-Wintner Theorem for a class of operators that includes compositions of ergodic Koopman operators with contractive multiplication operators. Our results are new even in the case of complex-valued functions, as they also apply to some non-positive non-contractive operators, and they give new uniform pointwise theorems for ergodic, weakly mixing, and mildly mixing Koopman operators.

math.FA↗

arXiv2024

Van der Corput's difference theorem for amenable groups and the left regular representation

We establish a connection between two variants of van der Corput's Difference Theorem (vdCDT) for countably infinite amenable groups $G$ and the ergodic hierarchy of mixing properties of a unitary representation $U$ of $G$. In particular, we show that one variant of vdCDT corresponds to subrepresentations of the left regular representation, and another variant of vdCDT corresponds to the absence of finite dimensional subrepresentations. We then obtain applications for measure preserving actions of countably infinite abelian groups.

math.DS↗

arXiv2024

Koopman representations for positive definite functions

We show that for any locally compact second countable group $G$ and any continuous positive definite function $ϕ:G\rightarrow\mathbb{C}$, there exists an ergodic measure preserving system $(X,\mathscr{B},μ,\{T_g\}_{g \in G})$ and a function $f \in L^2(X,μ)$ for which $ϕ(g) = \langle T_gf,f\rangle$. We also show that if $G$ is a countably infinite abelian group, then there exists a (not necessarily ergodic) measure preserving system $(X,\mathscr{B},μ,\{T_g\}_{g \in G})$ and a function $f \in L^2(X,μ)$ with $|f| = ϕ(0)$ and $ϕ(g) = \langle T_gf,f\rangle$.

math.GR↗

arXiv2024

A generalization of van der Corput's Difference Theorem

We prove a generalization of van der Corput's Difference Theorem in the theory of uniform distribution by establishing a connection with unitary operators that have Lebesgue spectrum. This allows us to show, for example, that if $(x_n)_{n = 1}^\infty \subseteq [0,1]$ is such that $(x_{n+h}-x_n)_{n = 1}^\infty$ is uniformly distributed for all $h \in \mathbb{N}$, then $(x_{n_k})_{k = 1}^\infty$ is uniformly distributed, where $(n_k)_{k = 1}^\infty$ is an enumeration of the $1s$ in the classical Thue-Morse sequence. We also establish a variant of van der Corput's Difference Theorem that is connected to unitary operators with continuous spectrum. Lastly, we obtain a new characterization of those sequence $(x_n)_{n = 1}^\infty \subseteq [0,1]$ for which $(x_{n+h},x_n)_{n = 1}^\infty$ is uniformly distributed in $[0,1]^2$ for all $h \in \mathbb{N}$.

math.DS↗

arXiv2023

A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.

math.DS↗

arXiv2023

On The Partition Regularity of $ax+by = cw^mz^n$

Csikvári, Gyarmati, and Sárközy showed that the equation $x+y = z^2$ is not partition regular (PR) over $\mathbb{N}$ and asked if the equation $x+y = wz$ is PR over $\mathbb{N}$. Bergelson and Hindman independently answered this question in the positive. We generalize this result by giving a partial classification of the $a,b,c \in \mathbb{Z}\setminus\{0\}$ and $m,n \in \mathbb{N}$ for which the equation $ax+by = cw^mz^n$ is PR over $\mathbb{Z}\setminus\{0\}$. We show that if $m,n \ge 2$, then $ax+by = cw^mz^n$ is PR over $\mathbb{Z}\setminus\{0\}$ if and only if $a+b = 0$. Next, we show that if $n$ is odd, then the equation $ax+by = cwz^n$ is PR over $\mathbb{Z}\setminus\{0\}$ if and only if one of $\frac{a}{c}, \frac{b}{c},$ or $\frac{a+b}{c}$ is an $n$th power in $\mathbb{Q}$. We come close to a similar characterization of the partition regularity of $ax+by = cwz^n$ over $\mathbb{Z}\setminus\{0\}$ for even $n$, and we examine some equations whose partition regularity remain unknown, such as $16x+17y = wz^8$. In order to show that the equation $ax+by = cwz^n$ is not PR over $\mathbb{Z}\setminus\{0\}$ for certain values of $a,b,c,$ and $n$, we prove a partial generalization of the criteria of Grunwald and Wang for when $α\in \mathbb{Z}$ is an $n$th power modulo every prime $p$. In particular, we show that for any odd $n$ and any $α,β,γ\in \mathbb{Q}$ that are not $n$th powers, there exist infinitely many primes $p \in \mathbb{N}$ for which none of $α,β,$ and $γ$ are $n$th powers modulo $p$. Similarly, we show that for any even $n$ and any $α,β,γ\in \mathbb{Q}$ that are not $\frac{n}{2}$th powers, with one not an $\frac{n}{4}$th power if $4|n$, there exist infinitely many primes $p \in \mathbb{N}$ for which $α,β,$ and $γ$ are not $n$th powers modulo $p$. Part of the abstract was removed here.

math.CO↗

arXiv2021

Pointwise Ergodic Theorems for Higher Levels of Mixing

We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly mixing measure preserving systems. We show that our pointwise theorem for weakly mixing systems is strictly stronger than the Wiener-Wintner Theorem. We also show that our pointwise Theorems for weakly mixing and strongly mixing systems characterize weakly mixing systems and strongly mixing systems respectively. The methods of this paper also allow one to prove an enhanced pointwise ergodic theorem for other levels of the ergodic hierarchy such as ergodicity and mild mixing but not K-mixing. The author plans to include these additional pointwise ergodic theorems in his thesis.

math.DS↗