arXiv subjects

Alexander Fish

Publications and source records attributed to Alexander Fish.

At least 19 recordsLinked to original sources

arXiv2026

Autonomous Collaborative Learning Among an Ensemble of Tsetlin Machines with Consensus-Based Inference

Tsetlin Machine (TM) is a rule-based machine-learning algorithm comprising collectives of two-action Tsetlin Automata (TAs) that cooperatively form conjunctive logical clauses from Boolean inputs through stochastic feedback. Although few recent studies have examined TM Federated Learning, the broader area of distributed and decentralized TM learning has not received much attention in the existing literature and warrants further exploration. In this work, we propose a paradigm for decentralized collaborative learning under a vertical feature-partitioning setting among an ensemble of Tsetlin Machines using consensus-based inference. Within this decentralized paradigm, each agent maintains its own private TM model, and there is no exchange of raw data among agents. Inference combines individual agents model predictions into a global consensus. The paradigm accommodates heterogeneous TM-based agents with differing data acquisition means, local data distributions, or computational resources, thereby facilitating the integration and fusion of information in settings such as multi-modal sensing environments. Experiments conducted using two-dimensional grid and connected graph network topologies demonstrate that the classification accuracies achieved are comparable to those of centralized models.

cs.LG↗

arXiv2026

Directional expansion in ergodic actions of countable groups

We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every totally ergodic unitary representation has arbitrarily small fixed-vector projections along infinite cyclic subgroups. This property implies directional expansivity for all totally ergodic actions. We prove that all infinite finitely generated nilpotent groups have the cyclic escape property, and conjecture the same for all infinite finitely generated polycyclic groups. We also prove the cyclic escape property for higher-rank simple lattices whose finite-dimensional unitary representations all have finite image; in particular, for $SL_n(\mathbb Z)$, $PSL_n(\mathbb Z)$, and $PGL_n(\mathbb Z)$, $n\geq 3$. By contrast, free groups of rank at least two do not have the cyclic escape property. The proofs exhibit two independent mechanisms: central spectral structure in nilpotent groups and stationary character rigidity in higher-rank lattices.

math.GR↗

arXiv2026

Trace spectra of simplices in large sets

Given an ordered tuple $\mathbf v=(v_0,\ldots,v_d)$ of vectors in $\mathbb{R}^d$, let $A_{\mathbf v}=[\,v_1-v_0\ \cdots\ v_d-v_0\,]$ be its edge matrix. We prove that, in every finite colouring of $\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \[ (c_2(A_{\mathbf v}),\ldots,c_d(A_{\mathbf v})). \] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\mathbf v})=\det(A_{\mathbf v})$. We also prove a discrete analogue: if $E\subseteq\mathbb{Z}^d$ has positive upper Banach density, then, for some $q\geq 1$, the set of coefficient tuples realized by ordered tuples in $E$ contains \[ q^2\mathbb{Z}\times q^3\mathbb{Z}\times\cdots\times q^d\mathbb{Z}. \] Finally, we show that the ordinary trace $c_1(A_{\mathbf v})$ cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices.

math.DS↗

arXiv2025

Dynamics of multiplicative groups over fields and Folner-Kloosterman sums

For two countably infinite fields whose multiplicative groups are isomorphic, we examine invariant couplings between the actions that these groups induce on the additive Pontryagin duals of the fields. We show that the actions are disjoint unless the fields themselves are isomorphic and the group isomorphism extends (possibly after a finite twist) to a field isomorphism. As an application, we establish equidistribution of F\o lner-Kloosterman sums - an extension of classical Kloosterman sums to infinite fields. Unlike the classical case over algebraic closures of finite fields, these averages exhibit an inherent multiplicative asymmetry, revealing new and fundamentally different behavior. Finally, we derive several combinatorial consequences, including results on sum-product phenomena and a Furstenberg--S\'ark\"ozy-type theorem for Laurent polynomials over general fields.

math.DS↗

arXiv2025

Ehrhart spectra of large subsets of $\mathbb{Z}^r$

This paper introduces and studies the Ehrhart spectrum of a set $E \subseteq \mathbb{Z}^r$, defined as the set of all Ehrhart polynomials of simplices with vertices in $E$, generalizing the notion of volume spectrum. We show that for any $E \subseteq \mathbb{Z}^r$ with positive upper Banach density, there is some $n \in \mathbb{Z}^r$ such that the Ehrhart spectrum of $n \mathbb{Z}^r$ is contained in the Ehrhart spectrum of $E$, generalizing an earlier result by the first and third author for the volume spectrum of $E$.

math.DS↗

arXiv2024

Toward an Ion-Based Large-Scale Integrated Circuit: Circuit Level Design, Simulation, and Integration of Iontronic Components

Iontronics combines ions as charge carriers with electronic-like operations, enabling unique information processing, chemical regulation, and enhanced bio-integrability. Standard simulation tools encounter difficulties in effectively modeling the behavior of integrated iontronic components, highlighting the need for specialized design and simulation approaches. This paper presents a design methodology for iontronic integrated circuits, inspired by well-established electronic design methodologies and made possible by the development of a compact model for the iontronic bipolar diode. Grounded in the diode's physical properties and observed behavior, this model provides a conceptual framework that could be applied to other iontronic components. It is implemented using standard VLSI (Very Large-Scale Integration) electronic design tools, enabling simulations that demonstrate diode-based iontronic circuit behaviors and laying the groundwork for the design and simulation of hybrid systems integrating electronic and iontronic circuits. The proposed iontronic circuit simulation approach enables the exploration of how component uniformity influences circuit behavior, as well as the impact of diode parameters and a deeper understanding of diode characteristics from a circuit perspective. These insights are expected to contribute to the development of more complex and efficient iontronic circuits, bringing us closer to practical and groundbreaking applications in the field.

cs.ET↗

arXiv2024

Quantitative expansivity for ergodic $\mathbb{Z}^d$ actions

We study expansiveness properties of positive measure subsets of ergodic $\mathbb{Z}^d$-actions along two different types of structured subsets of $\mathbb{Z}^d$, namely, cyclic subgroups and images of integer polynomials. We prove quantitative expansiveness properties in both cases and strengthen combinatorial results obtained by Bj\"orklund and Fish in arXiv:2401.03724, and Bulinski and Fish in arXiv:2102.05862. Our methods unify and strengthen earlier approaches used in arXiv:2401.03724 and arXiv:2102.05862 and to our surprise, also yield a counterexample to a certain pinned variant of the polynomial Bogolyubov theorem.

math.DS↗

arXiv2024

An inverse of Furstenberg's correspondence principle and applications to nice recurrence

We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,\mu,T)$ and $A\subset X$ measurable there exists a set $E \subset \mathbb{N}$ such that \[ \mu\left( \bigcap_{i=1}^k T^{-n_i}A\right) = \lim_{N\to \infty} \frac{\left|\left( \bigcap_{i=1}^k (E-n_i) \right)\cap \{0,\dots,N-1\}\right|}{N}\] for all $k,n_1,\dots,n_k \in \mathbb{N}$. As a corollary we show that a set $R\subset \mathbb{N}$ is a set of nice recurrence if and only if it is nicely intersective. Together, the inverse of Furstenberg's correspondence principle and it's corollary partially answer two questions of Moreira.

math.DS↗

arXiv2024

A Szemer\'edi type theorem for sets of positive density in approximate lattices

An extension of Szemer\'edi's Theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Via a novel version of Furstenberg's Correspondence principle, which should be of independent interest, we show that our Szemer\'edi Theorems can be deduced from a general \emph{transverse} multiple recurrence theorem, which we establish using recent works of Austin.

math.DS↗

arXiv2024

Simplices in large sets and directional expansion in ergodic actions

In this paper we study ergodic $\mathbb{Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all $r$-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham.

math.DS↗

arXiv2023

On-chip fully reconfigurable Artificial Neural Network in 16 nm FinFET for Positron Emission Tomography

Smarty is a fully-reconfigurable on-chip feed-forward artificial neural network (ANN) with ten integrated time-to-digital converters (TDCs) designed in a 16 nm FinFET CMOS technology node. The integration of TDCs together with an ANN aims to reduce system complexity and minimize data throughput requirements in positron emission tomography (PET) applications. The TDCs have an average LSB of 53.5 ps. The ANN is fully reconfigurable, the user being able to change its topology as desired within a set of constraints. The chip can execute 363 MOPS with a maximum power consumption of 1.9 mW, for an efficiency of 190 GOPS/W. The system performance was tested in a coincidence measurement setup interfacing Smarty with two groups of five 4 mm x 4 mm analog silicon photomultipliers (A-SiPMs) used as inputs for the TDCs. The ANN successfully distinguished between six different positions of a radioactive source placed between the two photodetector arrays by solely using the TDC timestamps.

eess.SY↗

arXiv2022

Glasner property for linear group actions and their products

A theorem of Glasner from 1979 shows that if $Y \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $\epsilon > 0$ there exists an integer $n$ such that $nY$ is $\epsilon$-dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on $\mathbb{T}^d$ also satisfy such a \textit{Glasner property} where each infinite set (in fact, arbitrarily large finite set) will have an $\epsilon$-dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski-connected Zariski-closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.

math.DS↗

arXiv2021

Arithmetic subtrees in large subsets of products of trees

Furstenberg-Weiss have extended Szemer\'edi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend the multidimensional Szemer\'edi theorem by demonstrating the existence of certain arithmetic structures in large subsets of a cartesian product of trees.

math.CO↗

arXiv2021

Reconstructing a minimal topological dynamical system from a set of return times

We investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations this is indeed the case as long as the closure of this open set has no non-trivial translational symmetries. For instance, we show that under this assumption two Kronecker systems with the same set of return times must be isomorphic. More generally, we show that if a minimal dynamical system has a set of return times that coincides with a set of return times to some open set in a Kronecker system with translationarily asymmetric closure, then that Kronecker system must be a factor. We also study similar problems involving Nilsystems and polynomial return times. We state a number of questions on whether these results extend to other homogeneous spaces and transitive group actions, some of which are already interesting for finite groups.

math.DS↗

arXiv2021

Glasner property for unipotently generated group actions on tori

A theorem of Glasner from 1979 shows that if $A \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $\epsilon > 0$ there exists an integer $n$ such that $nA$ is $\epsilon$-dense and Berend-Peres later showed that in fact one can take $n$ to be of the form $f(m)$ for any non-constant $f(x) \in \mathbb{Z}[x]$. Alon and Peres provided a general framework for this problem that has been used by Kelly-L\^{e} and Dong to show that the same property holds for various linear actions on $\mathbb{T}^d$. We complement the result of Kelly-L\^{e} on the $\epsilon$-dense images of integer polynomial matrices in some subtorus of $\mathbb{T}^d$ by classifying those integer polynomial matrices that have the Glasner property in the full torus $\mathbb{T}^d$. We also extend a recent result of Dong by showing that if $\Gamma \leq \operatorname{SL}_d(\mathbb{Z})$ is generated by finitely many unipotents and acts irreducibly on $\mathbb{R}^d$ then the action $\Gamma \curvearrowright \mathbb{T}^d$ has a uniform Glasner property.

math.DS↗

arXiv2021

On almost Cap sets in three variables and the multivariable Cap set problem

In this note we prove that almost cap sets $A \subset \mathbb{F}_q^n$, i.e., the subsets of $\mathbb{F}_q^n$ that do not contain too many arithmetic progressions of length three, satisfy that $|A| < c_q^n$ for some $c_q < q$. As a corollary we prove a multivariable analogue of Ellenberg-Gijswijt theorem.

math.NT↗

arXiv2021

Quantitative twisted patterns in positive density subsets

We make quantitative improvements to recently obtained results on the structure of the image of a large difference set under certain quadratic forms and other homogeneous polynomials. Previous proofs used deep results of Benoist-Quint on random walks in certain subgroups of $\operatorname{SL}_r(\mathbb{Z})$ (the symmetry groups of these quadratic forms) that were not of a quantitative nature. Our new observation relies on noticing that rather than studying random walks, one can obtain more quantitative results by considering polynomial orbits of these group actions that are not contained in cosets of submodules of $\mathbb{Z}^r$ of small index. Our main new technical tool is a uniform Furstenberg-S\'{a}rk\"{o}zy theorem that holds for a large class of polynomials not necessarily vanishing at zero, which may be of independent interest and is derived from a density increment argument and Hua's bound on polynomial exponential sums.

math.DS↗

arXiv2020

Direct and inverse results for popular differences in trees of positive dimension

We establish analogues for trees of results relating the density of a set $E \subset \mathbb{N}$, the density of its set of popular differences, and the structure of $E$. To obtain our results, we formalise a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting we use a recent result of the first author with Bj\"orklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.

math.DS↗