arXiv subjects

Anh N. Le

Publications and source records attributed to Anh N. Le.

17 recordsLinked to original sources

arXiv2026

Bohr sets in sumsets III: expanding difference sets and almost Bohr sets

Let $G$ be a discrete abelian group. Følner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach density. We study the sets $S \subseteq G$ for which $A - A + S$ contains a Bohr set for every $A \subseteq G$ of positive upper Banach density. For $G = \mathbb{Z}$, we show that the sets $\{n^2: n \in \mathbb{N}\}$, $\{p - 1: p \text{ prime}\}$, and $\{ \lfloor n^c \rfloor: n \in \mathbb{N} \}$ with $c > 0$, have this property. Moreover, we prove that there are sets $A, B \subseteq \mathbb{Z}$ such that $A$ is dense in the Bohr topology of $\mathbb{Z}$, $d^*(B) > 0$, while $A + B$ is not piecewise Bohr, answering two questions of the second author in [31]. We also study those sets $S$ such that $A + S$ contains a Bohr set for every almost Bohr set $A$. As applications, we prove: (i) If $ϕ_1, ϕ_2: G \to G$ are (not necessarily commuting) homomorphisms with finite indices $[G: ϕ_i(G)]$, and $C \subseteq G$ is a central set, then $ϕ_1(C) - ϕ_1(C) + ϕ_2(C)$ contains a Bohr set. This answers one of our questions in [35] and generalizes results in [44, 48]; (ii) Every set of pointwise recurrence in $\mathbb{Z}$ is a set of nice recurrence and a van der Corput set, extending known properties of sets of pointwise recurrence studied in [26, 27, 40].

math.DS↗

arXiv2026

A structure theorem for polynomial return-time sets in minimal systems

We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent.

math.DS↗

arXiv2026

On sets of pointwise recurrence and dynamically thick sets

A set $A \subseteq \mathbb{N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X, T)$, all $x \in X$, and all open neighborhoods $U \subseteq X$ of $x$, there exists a time $n \in A$ such that $T^n x \in U$. The set $A$ is dynamically thick if the same holds for all non-empty, open sets $U \subseteq X$. Our main results give combinatorial characterizations of sets of pointwise recurrence and dynamically thick sets that allow us to answer questions of Host, Kra, Maass and Glasner, Tsankov, Weiss, and Zucker. We also introduce and study a local version of dynamical thickness called dynamical piecewise syndeticity. We show that dynamically piecewise syndetic sets are piecewise syndetic, generalizing results of Dong, Glasner, Huang, Shao, Weiss, and Ye. The proofs involve the algebra of families of large sets, dynamics on the space of ultrafilters, and our recent characterization of dynamically syndetic sets.

math.DS↗

arXiv2025

Bohr sets in sumsets I: Compact abelian groups

Let $G$ be a compact abelian group and $ϕ_1, ϕ_2, ϕ_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $ϕ_i$'s, we prove the existence of Bohr sets in the sumset $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$, where $A$ is either a set of positive Haar measure, or comes from a finite partition of $G$. The first result generalizes theorems of Bogolyubov and Bergelson-Ruzsa. As a variant of the second result, we show that for any partition $\mathbb{Z} = \bigcup_{i=1}^r A_i$, there exists an $i$ such that $A_i - A_i + sA_i$ contains a Bohr set for any $s \in \mathbb{Z} \setminus \{ 0 \}$. The latter is a step toward an open question of Katznelson and Ruzsa.

math.CO↗

arXiv2025

Dynamically syndetic sets and the combinatorics of syndetic, idempotent filters

A subset of the positive integers is dynamically central syndetic if it contains the times that a point returns to a neighborhood of itself in a minimal topological dynamical system. These sets are part of the highly-influential link between dynamics and combinatorics forged by Furstenberg and Weiss in the 1970's. Our main result is a characterization of dynamically central syndetic sets as precisely those sets that belong to syndetic, idempotent filters. This gives a "global" analogue to the well-known "local" characterization of Furstenberg's central sets as members of minimal, idempotent ultrafilters. Applying the main result, we answer two open questions posed by Host, Kra, and Maass concerning sets of pointwise topological recurrence.

math.DS↗

arXiv2025

On subshifts with low maximal pattern complexity

For a finite alphabet $\mathcal{A}$ and a sequence $x \in \mathcal{A}^{\mathbb{N}}$, Kamae and Zamboni defined the maximal pattern complexity function $p^*_x(n)$ as a natural generalization of usual word complexity. They defined a nonperiodic sequence $x$ to be pattern Sturmian if it achieves the minimal growth rate $p^*_x(n) = 2n$, and asked the question of whether one could classify recurrent pattern Sturmian sequences. We answer their question by characterizing recurrent pattern Sturmian sequences as one of two known types: either a coding of an irrational circle rotation by two intervals, or an element of what we call a nearly simple Toeplitz subshift. We also show that nonrecurrent pattern Sturmian sequences are either very close to constant (such examples were given by Kamae and Zamboni) or a (nonrecurrent) coding of an irrational circle rotation by two intervals. Our main new technique is to use topological properties of the maximal equicontinuous factor (MEF) of the subshift generated by $x$. In this way, we prove a general structural result about sequences with non-superlinear maximal pattern complexity: they are either nonrecurrent or minimal with MEF either an odometer or the product of a circle with a finite cyclic group.

math.DS↗

arXiv2024

Interpolation sets for dynamical systems

Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s. A set $S \subset \mathbb{N}$ is an interpolation set for a class of topological dynamical systems $\mathcal{C}$ if any bounded sequence on $S$ can be extended to a sequence that arises from a system in $\mathcal{C}$. In this paper, we provide combinatorial characterizations of interpolation sets for: $\bullet$ (totally) minimal systems; $\bullet$ topologically (weak) mixing systems; $\bullet$ strictly ergodic systems; and $\bullet$ zero entropy systems. Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets.

math.DS↗

arXiv2024

Averages of completely multiplicative functions over the Gaussian integers -- a dynamical approach

We prove a pointwise convergence result for additive ergodic averages associated with certain multiplicative actions of the Gaussian integers. We derive several applications in dynamics and number theory, including: (i) Wirsing's theorem for Gaussian integers: if $f\colon \mathbb{G} \to \mathbb{R}$ is a bounded completely multiplicative function, then the following limit exists: $$\lim_{N \to \infty} \frac{1}{N^2} \sum_{1 \leq m, n \leq N} f(m + {\rm i} n).$$ (ii) An answer to a special case of a question of Frantzikinakis and Host: for any completely multiplicative real-valued function $f: \mathbb{N} \to \mathbb{R}$, the following limit exists: $$\lim_{N \to \infty} \frac{1}{N^2} \sum_{1 \leq m, n \leq N} f(m^2 + n^2).$$ (iii) A variant of a theorem of Bergelson and Richter on ergodic averages along the $Ω$ function: if $(X,T)$ is a uniquely ergodic system with unique invariant measure $μ$, then for any $x\in X$ and $f\in C(X)$, $$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{Ω(m^2 + n^2)}x)=\int_Xf \ dμ.$$

math.DS↗

arXiv2024

Intersective sets for sparse sets of integers

For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is $E$-intersective if for every $A \subset E$ having positive upper relative density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When $E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic intersectivity. In this article, we investigate to which extent known intersectivity results hold in the relative setting when $E = \mathbb{P}$, the set of primes, or other sparse subsets of $\mathbb{N}$. Among other things, we prove: -There exists an intersective set that is not $\mathbb{P}$-intersective. -However, every $\mathbb{P}$-intersective set is intersective. -There exists a chromatically $\mathbb{P}$-intersective set which is not intersective (and therefore not $\mathbb{P}$-intersective). -The set of shifted Chen primes $\mathbb{P}_{\mathrm{Chen}} + 1$ is $\mathbb{P}$-intersective (and therefore intersective).

math.NT↗

arXiv2023

Bohr sets in sumsets II: countable abelian groups

We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting $G$ be a countable discrete abelian group and $ϕ_1, ϕ_2, ϕ_3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that (1) If $A \subset G$ has positive upper Banach density and $ϕ_1 + ϕ_2 + ϕ_3 = 0$, then $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in $\mathbb{Z}$ and a recent result of the first author. (2) For any partition $G = \bigcup_{i=1}^r A_i$, there exists an $i \in \{1, \ldots, r\}$ such that $ϕ_1(A_i) + ϕ_2(A_i) - ϕ_2(A_i)$ contains a Bohr set. This generalizes a result of the second and third authors from $\mathbb{Z}$ to countable abelian groups. (3) If $B, C \subset G$ have positive upper Banach density and $G = \bigcup_{i=1}^r A_i$ is a partition, $B + C + A_i$ contains a Bohr set for some $i \in \{1, \ldots, r\}$. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. These results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices $[G:ϕ_j(G)]$, the upper Banach density of $A$ (in (1)), or the number of sets in the given partition (in (2) and (3)).

math.CO↗

arXiv2022

Additive averages of multiplicative correlation sequences and applications

We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form $\{(an+b)^{\ell}/(cn+d)^{\ell}: n \in \mathbb{N}\}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.

math.DS↗

arXiv2021

A decomposition of multicorrelation sequences for commuting transformations along primes

We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $ε>0$, as the sum of a nilsequence $ϕ(n)$ and a sequence $ω(n)$ satisfying $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N |ω(n)|<ε$ and $\lim_{N\to\infty}\frac{1}{|\mathbb{P}\cap [N]|}\sum_{p\in \mathbb{P}\cap [N]} |ω(p)|<ε$.

math.DS↗

arXiv2021

Sublacunary sets and interpolation sets for nilsequences

A set $E \subset \mathbb{N}$ is an interpolation set for nilsequences if every bounded function on $E$ can be extended to a nilsequence on $\mathbb{N}$. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as the result, obtain a new example of interpolation set for $2$-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.

math.DS↗

arXiv2020

Structure of multicorrelation sequences with integer part polynomial iterates along primes

Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability space $(X,{\mathcal B},μ),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $ε> 0$ and multicorrelation sequences of the form $\displaystyleα(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;dμ$ we show that there exists a nilsequence $ψ$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |α(n) - ψ(n)| \leq ε$ and $\displaystyle\lim_{N \to \infty} \frac{1}{π(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |α(p) - ψ(p)| \leq ε.$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13].

math.DS↗

arXiv2019

Optimal lower bounds for multiple recurrence

Let $(X, \mathcal{B},μ,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $ε>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colonμ(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> μ(A)^{k+1} - ε\right\} \end{split} \end{equation*} for various families $\{f_1,\dots,f_k\}$ of sequences $f_i\colon \mathbb{N} \to \mathbb{N}$. For $k \leq 3$ and $f_{i}(n)=i f(n)$, 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.

math.DS↗

arXiv2019

Interpolation sets and nilsequences

To give positive answer to a question of Frantzikinakis, we study a class of subsets of $\mathbb{N}$, called interpolation sets, on which every bounded sequence can be extended to an almost periodic sequence on $\mathbb{N}$. Strzelecki has proved that lacunary sets are interpolation sets. We prove that sets that are denser than all lacunary sets cannot be interpolation sets. We also extend the notion of interpolation sets to nilsequences and show that the analogue to Frantzikinakis' question for arbitrary sequences is false.

math.DS↗

arXiv2012

Analyzing Nonblocking Switching Networks using Linear Programming (Duality)

The main task in analyzing a switching network design (including circuit-, multirate-, and photonic-switching) is to determine the minimum number of some switching components so that the design is non-blocking in some sense (e.g., strict- or wide-sense). We show that, in many cases, this task can be accomplished with a simple two-step strategy: (1) formulate a linear program whose optimum value is a bound for the minimum number we are seeking, and (2) specify a solution to the dual program, whose objective value by weak duality immediately yields a sufficient condition for the design to be non-blocking. We illustrate this technique through a variety of examples, ranging from circuit to multirate to photonic switching, from unicast to $f$-cast and multicast, and from strict- to wide-sense non-blocking. The switching architectures in the examples are of Clos-type and Banyan-type, which are the two most popular architectural choices for designing non-blocking switching networks. To prove the result in the multirate Clos network case, we formulate a new problem called {\sc dynamic weighted edge coloring} which generalizes the {\sc dynamic bin packing} problem. We then design an algorithm with competitive ratio 5.6355 for the problem. The algorithm is analyzed using the linear programming technique. A new upper-bound for multirate wide-sense non-blocking Clos networks follow, improving upon a decade-old bound on the same problem.

cs.DM↗