arXiv subjects

Dimitrios Charamaras

Publications and source records attributed to Dimitrios Charamaras.

6 recordsLinked to original sources

arXiv2026

Multiple ergodic averages for commuting multiplicative actions

We study the convergence of multiple ergodic averages involving several commuting multiplicative actions. We prove that for finitely generated systems, the averages $$\frac{1}{N}\sum_{n=1}^N S_{1,n}F_1\cdot\ldots\cdot S_{\ell,n}F_\ell$$ converge in norm, settling a conjecture of Frantzikinakis. Our methods rely on a novel generalization of K\'atai's orthogonality criterion that allows us to obtain box seminorm control, the machinery of magic extensions originating in the work of Host, and a delicate induction on the complexity of the initial averages that ultimately reduces our problem to well-known mean convergence results.

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 F{\o}lner 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 F{\o}lner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.

math.DS↗

arXiv2024

Asymptotic independence of $\Omega(n)$ and $\Omega(n+1)$ along logarithmic averages

Let $\Omega(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N \frac{a(\Omega(n))b(\Omega(n+1))}{n} = \Bigg(\frac{1}{N}\sum_{n=1}^N a(\Omega(n))\Bigg)\Bigg(\frac{1}{N}\sum_{n=1}^N b(\Omega(n))\Bigg) + \mathrm{o}_{N\to\infty}(1).$$ This generalizes a theorem of Tao on the logarithmically averaged two-point correlation Chowla conjecture. Our result is quantitative and the explicit error term that we obtain establishes double-logarithmic savings. As an application, we obtain new results about the distribution of $\Omega(p+1)$ as $p$ ranges over $\ell$-almost primes for a "typical" value of $\ell$.

math.NT↗

arXiv2024

On multiplicative recurrence along linear patterns

In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for actions of the multiplicative semigroup $(\mathbb{N}, \times)$ and provided some sufficient conditions for sets of the form $S=\{(an+b)/(cn+d) \colon n \in \mathbb{N} \}$ to be sets of recurrence for such actions. A necessary condition for $S$ to be a set of multiplicative recurrence is that for every completely multiplicative function $f$ taking values on the unit circle, we have that $\liminf_{n \to \infty} |f(an+b)-f(cn+d)|=0.$ In this article, we fully characterize the integer quadruples $(a,b,c,d)$ which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel concerning the pair $(n,n+1)$, as well as some results of Donoso, Le, Moreira and Sun. In addition, we prove that, under the same conditions on $(a,b,c,d)$, the set $S$ is a set of recurrence for finitely generated actions of $(\mathbb{N}, \times)$.

math.NT↗

arXiv2024

Mean value theorems in multiplicative systems and joint ergodicity of additive and multiplicative actions

In this paper we are concerned with the study of additive ergodic averages in multiplicative systems and the investigation of the "pretentious" dynamical behaviour of these systems. We prove a mean ergodic theorem (Theorem A) that generalises Hal\'asz's mean value theorem for finitely generated multiplicative functions taking values in the unit circle. In addition, we obtain two structural results concerning the "pretentious" dynamical behaviour of finitely generated multiplicative systems. Moreover, motivated by the independence principle between additive and multiplicative structures of the integers, we explore the joint ergodicity (as a natural notion of independence) of an additive and a finitely generated multiplicative action, both acting on the same probability space. In Theorem B, we show that such actions are jointly ergodic whenever no "local obstructions" arise, and we give a concrete description of these "local obstructions". As an application, we obtain some new combinatorial results regarding arithmetic configurations in large sets of integers including refinements of a special case of Szemer\'edi's theorem.

math.DS↗

arXiv2024

Finding product sets in some classes of amenable groups

In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erd\H{o}s about sumsets in large subsets of the natural numbers. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups, and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group $G$ from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in\mathbb{N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in a recent survey of Kra, Moreira, Richter and Robertson.

math.DS↗