Keivan Mallahi-Karai

Constructor University

We prove a generalization of a theorem of Eskin-Margulis-Mozes in an $S$-arithmetic setup: suppose that we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place, an irrational isotropic form ${\mathbf q}$ of rank $n\geq 4$ on $\mathbb{Q}_S$, a product of $p$-adic intervals ${\mathbb{I}}_p$, and a product $\Omega$ o...

Let $\mathbb{K}$ be the field of real or $p$-adic numbers, and $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with coefficients in $\mathbb{K}$. Employing ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will...

We study an extreme value distribution for the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed...

We define and study a new compactification, called the height compactification of the horospheric product of two infinite trees. We will provide a complete description of this compactification. In particular, we show that this compactification is isomorphic to the Busemann compactification when all the vertices of both trees have degrees of at leas...

Given a finite group G \mathrm {G} , the faithful dimension of G \mathrm {G} over C \mathbb {C} , denoted by m f a i t h f u l ( G ) m_\mathrm {faithful}(\mathrm {G}) , is the smallest integer n n such that G \mathrm {G} can be embedded in G L n ( C ) \mathrm {GL}_n(\mathbb {C}) . Continuing the work initiated by Bardestani et al. [Compos. Math. 15...

A countable group G is called k-linear sofic (for some 0 <k \le 1) if finite subsets of G admit "approximate representations" by complex invertible matrices in the normalized rank metric, so that non-identity elements are k-away from the identity. This class of groups was systematically studied by Arzhantseva and Paunescu [AP17], where it is shown...

We study the joint distribution of values of a pair consisting of a quadratic form $q$ and a linear form $\mathbf l$ over the set of integral vectors, a problem initiated by Dani-Margulis (1989). In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$ th...

This work is a continuation of Mallahi-Karai and Diederich (2019), where the authors introduced and studied the cube model as a multi-dimensional extension of the diffusion model in binary choice model. The aim of this note is to introduce and study the disk model, which can be viewed as a variation of the model introduced in the aforementioned pap...

In this work, we will introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls and provide several examples of such groups. In particular, this leads to new examples of groups s...

We propose a family of new multi-episode models of decision making with more than two alternatives. These models can be viewed as multi-dimensional extensions of the standard diffusion model (i.e. Wiener process) with two alternatives introduced in Laming (1968), Link and Heath (1975), Ratcliff (1978) while at the same time, they incorporate Tversk...

In this paper, we will study the behavior of the space of positive harmonic functions associated with the random walk on a countable group under the change of probability measure by a stopping time. We show that this space remains unchanged after applying a uniformly bounded stopping time.

Siegel's paradox is a fundamental question in international finance about exchange rates for futures contracts and has puzzled many scholars for over forty years. The unorthodox approach presented in this article leads to an arbitrage-free solution which is invariant under currency re-denominations and is symmetric, as explained. We will also give...

In this paper we will give various examples of exponentially distorted subgroups in linear groups, including some new example of subgroups of $SL_n(\mathbb{Z}[x])$ for $n \ge 3$, and show how they can be used to construct symmetric-key cryptographic platforms.

Let $\mathrm{G}$ be a finite group. The faithful dimension of $\mathrm{G}$ is defined to be the smallest possible dimension for a faithful complex representation of $\mathrm{G}$. Aside from its intrinsic interest, the problem of determining the faithful dimension of $p$-groups is motivated by its connection to the theory of essential dimension. In...

An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository pape...

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QGQ, called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ V form an edge if and only if Q(v − w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous...

Let F be a non-Archimedean local field with the ring of integers O and the prime ideal p and let G = Gad (O=pⁿ) be the adjoint Chevalley group. Let mf(G) denote the smallest possible dimension of a faithful representation of G. Using the Stone- von Neumann theorem, we determine a lower bound for mf(G) which is asymptotically the same as the results...

We will give a criterion for the amenability of arbitrary locally finite trees. The criterion is based on the trimming operator which is defined on the space of trees. As an application, we obtain a necessary and sufficient condition for that amenability of Galton-Watson trees.

We will prove an S-arithmetic version of a theorem of Dani-Margulis on the convergence of ergodic averages of a given bounded continuous function, when the initial point is outside certain compact subsets of the singular set associated to the unipotent flow.

In this note, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and Gowers' mixing inequality for quasirandom groups, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of th...

Let a finite group \(G\) act transitively on a finite set \(X\). A subset \(S\subseteq G\) is said to be intersecting if for any \(s_1,s_2\in S\), the element \(s_1^{-1}s_2\) has a fixed point. The action is said to have the weak Erdős–Ko–Rado (EKR) property, if the cardinality of any intersecting set is at most \(|G|/|X|\). If, moreover, any maxim...

For a group $G$, we denote by $m_f(G)$, the smallest dimension of a faithful complex representation of $G$. Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the maximal ideal $\mathfrak{p}$. In this paper, we compute the precise value of $m_f(G)$ when $G$ is the Heisenberg group over $\mathcal{O}/\mathfrak{p}^n$....

Let $G=\mathrm{G}_{ad}\left(\mathbb{Z}/(p^n\mathbb{Z})\right)$ be the adjoint Chevalley group and let $m_f(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone--von Neumann theorem, we determine a lower bound for $m_f(G)$ which is asymptotically the same as the results of Landazuri, Seitz and Zalesskii for...

Let a finite group G acts transitively on a finite set X. A subset S of G is called an intersecting family if for any s_1,s_2 in S, the element s_1^{-1}s_2 has a fixed point. We say that this action has the weak Erdos-Ko-Rado property, if the size of any intersecting family is bounded above by |G|/|X|. The action has the strong Erdos-Ko-Rado proper...

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\SL_k(\Z{p^n})$ and $\Sp_{2k}(\Z{p^n})$. Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we est...

In this paper we will describe a framework that allows us to connect the problem of hedging a portfolio in finance to the existence of Pareto optimal allocations in economics. We will show the solvability of both problems is equivalent to the No Good Deals assumption. We will then analyze the case of co-monotone additive monetary utility functions...