arXiv subjects

Ioannis Kousek

Publications and source records attributed to Ioannis Kousek.

8 recordsLinked to original sources

arXiv2026

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

We prove a refinement of a recent structural result in topological dynamics due to Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson, who showed that in a minimal system, polynomial return-times differ by non-piecewise syndetic sets from those in its maximal infinite-step pronilfactor. In particular, we prove that the infinite-step pronilfactor can be replaced by the maximal $k$-step pronilfactor of the system, where $k$ depends only on the given polynomials, with $k-1$ being equal to the Host-Kra complexity of the polynomials.

math.DS↗

arXiv2026

Sharp density conditions for infinite $B+B$ sumsets in abelian groups

Motivated by recent results \cite{charamaras_kousek_mountakis_radic2025BBingroups} on infinite sumsets of the form $B+B=\{b_1+b_2:b_1,b_2\in B\}$ in large subsets of abelian groups, and an old problem of Owings \cite[Problem E2494]{Owing_problems} about the partition regularity of $B+B$ in $2$ colours, we show the following theorem. Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $D: g\mapsto g+g$ has finite kernel. Let also $Φ=(Φ_N)_{N}$ be any Folner sequence in $G$ and $Φ/2=(D^{-1}(Φ_N))_{N}$. Then, if $A\subset G$ is such that $d_Φ(A)+d_{Φ/2}(A)>1$, there is an infinite set $B\subset G$ and some $t\in G$ for which $t+B+B\subset A$. We prove that this result implies the main theorem in \cite{charamaras_kousek_mountakis_radic2025BBingroups}, and construct an example to show the reverse implication does not hold. Moreover, we show that our main theorem is optimal in a strong sense. Namely, for any countable abelian group $(G,+)$ with the aforementioned assumptions -- which are necessary -- there exists a Folner sequence $Φ$ and a set $A\subset G$ so that $d_Φ(A)+d_{Φ/2}(A)=1$, but there is no infinite set $B\subset G$ and $t\in G$ for which $t+B+B\subset A$.

math.CO↗

arXiv2026

Uniqueness of a topological Furstenberg system

Given a semigroup $G$ and a bounded function $f: G \to \mathbb{C}$, a topological Furstenberg system of $f$ is a topological dynamical system $\mathbb{X}=(X, (T_g)_{g \in G})$ that encodes the dynamical behaviour of $f$. We show that $\mathbb{X}$ is unique up to topological isomorphism, thus providing a topological analogue of the measurable case established by Bergelson and Ferré Moragues for amenable semigroups. We also provide necessary and sufficient conditions for subsets of a group to have isomorphic Furstenberg systems. In addition, we study sets with minimal Furstenberg systems and identify them as a special subclass of dynamically syndetic sets. Moreover, we use this notion to obtain a new characterization of sets of topological recurrence.

math.DS↗

arXiv2025

On density analogs of Hindman's finite sums theorem

For any set $A$ of natural numbers with positive upper Banach density, we show the existence of an infinite set $B$ and sequences $(t_k)_{k\in \mathbb{N}}, (s_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F \subset B + s_k, 1 \leq |F| \leq k \right\}\subset A-t_k$, for every $k\in \mathbb{N}$. This strengthens the density finite sums theorem of Kra, Moreira, Richter, and Robertson. We further show, given such a set $A$, the existence of an infinite set $B$ and a sequence $(t_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F \subset B, |F| = k \right\}\subset A-t_k$, for every $k\in \mathbb{N}$. As a corollary, we obtain a sequence $(B_n)_{n\in \mathbb{N}}$ of infinite sets of natural numbers such that $B_1+\cdots +B_k \subset A$, for every $k\in \mathbb{N}$. We also establish the optimality of our main theorems by providing counterexamples to potential further generalizations, and thereby addressing questions of the aforementioned authors in the context of density analogs to Hindman's finite sums theorem.

math.DS↗

arXiv2025

Infinite unrestricted sumsets in subsets of abelian groups with large density

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.

math.DS↗

arXiv2025

Asymmetric infinite sumsets in large sets of integers

We show that for any set $A \subset \mathbb{N}$ with positive upper density and any $\ell,m \in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text{and}\ b_1 1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.

math.DS↗

arXiv2024

Revisiting sums and products in countable and finite fields

We establish a polynomial ergodic theorem for actions of the affine group of a countable field $K$. As an application, we deduce--via a variant of Furstenberg's correspondence principle--that for fields of characteristic zero, any "large" set $E\subset K$ contains "many" patterns of the form $\{p(x)+y,xy\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a new finitistic variant of Bergelson's "colouring trick", show that for $r\in \mathbb{N}$ fixed, any $r-$colouring of a large enough finite field will contain monochromatic patterns of the form $\{x,p(x)+y,xy\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalisation of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic "colouring trick", we provide a conditional, elementary generalisation of Green and Sanders' $\{x,y,x+y,xy\}$ theorem.

math.CO↗

arXiv2024

Infinite unrestricted sumsets of the form $B+B$ in sets with large density

For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically, when the lower density $\underline{d}(A) >1/2$ or the upper density $\overline{d}(A)> 3/4$, the existence of such a set $B\subset \mathbb{N}$ and $t\in \{0,1\}$ is assured. Furthermore, whenever $\underline{d}(A) > 3/4$ or $\overline{d}(A)>5/6$, we show that the shift $t$ is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three-coloring of the natural numbers that does not contain a monochromatic $B+B+t$ for any infinite set $B \subset \mathbb{N}$ and number $t \in \mathbb{N}$.

math.DS↗