arXiv subjects

Keivan Mallahi-Karai

Publications and source records attributed to Keivan Mallahi-Karai.

At least 19 recordsLinked to original sources

arXiv2026

On the Dimension of the Best-Worst Choice Polytope

For a set of $n$ alternatives, the best--worst choice polytope is the convex hull of the deterministic best--worst choice vectors induced by all linear rankings of the alternatives. The question of determining the dimension of this polytope as a function of $n$ has been raised by Doignon \cite{Doignon2023}. In this paper we answer this question by finding an explicit formula for this dimension. Our proof is motivated by a representation-theoretic interpretation of the relevant rank problem for the symmetric group, but is presented entirely in elementary linear-algebraic terms. The connection with representation theory will be explored elsewhere.

math.CO↗

arXiv2026

An infinite family of trees with irreducible characteristic polynomials

Consider the tree obtained by attaching a leaf to the third vertex of a path with n-1 vertices. In this note, we prove that the characteristic polynomial of this tree is irreducible when n belongs to certain arithmetic progressions modulo 30. As a result there are infinitely many pairwise non-isomorphic trees with an irreducible the characteristic polynomial. Our proof combines several number-theoretic arguments with a result of Gross, Hironaka, and McMullen [GHM09] concerning the cyclotomic factors of the Coxeter polynomials associated with the diagrams En. This resolves affirmatively a conjecture of Akbari, Kumar, Mohar and Pragada.

math.CO↗

arXiv2026

Random free semigroups of affine groups

We investigate random freeness of semigroups in the solvable non-virtually nilpotent setting. We focus on a correlated affine model, namely semigroups generated by the two components of a finitely supported random walk $L_n=(L_{n,1},L_{n,2})$ on $\operatorname{Aff}(K)^2$ whose two components share a common linear part. In this model, we show that the long-term behavior of freeness is completely governed by an abelian shadow, namely the projected random walk on the multiplicative group $A\subset K^\times$ generated by the common multipliers. If this walk is transient, then {the semigroup} $\langle L_{n,1},L_{n,2}\rangle_+$ is eventually free almost surely. If it is recurrent, then freeness does not stabilize: the set of non-free times is almost surely infinite, yet has almost sure density zero. Moreover, the obstructions to freeness depends solely on the common linear part and admits an explicit arithmetic description in terms of roots of Littlewood polynomials. The proof combines a local contraction theorem for affine random walks over arbitrary local fields, developed in the appendix and related to the theory of critical affine random walks, with a ping-pong argument played at a time-dependent place.

math.PR↗

arXiv2026

Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$

Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the nilradical of $\mathfrak b_n$. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a$ of $\mathfrak u_n(\mathbb F_q)$, and every partition $μ$ of $n$, we prove two formulas for the number of elements of $\mathfrak a$ of Jordan type $μ$: the first one is the Hall scalar product of a modified Hall-Littlewood function indexed by $μ$ and a chromatic quasisymmetric function associated to $\mathfrak a$, and the second one is in terms of an explicit collection of standard tableaux. In the special case that $\mathfrak a$ is the nilradical $\mathfrak u_Λ(\mathbb F_q)$ of the parabolic subalgebra associated to a composition $Λ$ of $n$, our first formula reduces to a result of Karp and Thomas: up to an explicit polynomial factor in $q$, the number of elements in $\mathfrak u_Λ(\mathbb F_q)$ of Jordan type $μ$ is equal to the coefficient of the monomial $\mathsf x^Λ$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{μ'}(\mathsf x;q^{-1},t)$ at $t=0$. We give three applications: (1) a formula for the number of points of a nilpotent Hessenberg variety, (2) a formula for the number of $X\in \mathfrak u_Λ(\mathbb F_q)$ that satisfy $X^2=0$, which in the special case $Λ=(1^n)$ is different from the Kirillov-Melnikov-Ekhad-Zeilberger formula, and (3) a formula for the number of double cosets $\mathrm U_1\backslash\mathrm{GL}_n(\mathbb F_q)/\mathrm U_2$ where $\mathrm U_1$ and $\mathrm U_2$ are unipotent subgroups corresponding to two $\mathfrak b_n(\mathbb F_q)$-stable ideals.

math.CO↗

arXiv2024

Optimal Linear Sofic Approximations Of Countable Groups

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 that such a group is always 1/4-linear sofic. In this paper, we will study the optimality of this result for general countable groups and show that every linear sofic group is 1/2-linear sofic, and 1/2 cannot be improved. However, if G is assumed to be torsion-free, then it is 1-linear sofic. These results answer a question posed by G. Arzhantseva in her talk in the IAS Stability and Testability lecture series. We also study the optimal linear sofic constant of finite groups over C and fields of positive characteristic. For the proof, we establish effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest. Another ingredient of the proof involves bounding integrals of certain trigonometric sums.

math.GR↗

arXiv2024

On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$

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 analytic formulas for $F(r)$ for $r \in [-\frac{1}{2} \log 2, \infty)$, and establish asymptotic behavior of $F(r)$ as $r \to -\infty$.

math.DS↗

arXiv2024

Spectral Independence

We prove the spectral gap property for random walks on the product of two non-locally isomorphic analytic real or p-adic compact groups with simple Lie algebras, under the necessary condition that the marginals posses a spectral gap. Furthermore, we give additional control on the spectral gap depending on certain specific properties of the given groups and marginals; in particular, we prove some new cases of the super-approximation conjecture. One ingredient of the proof is a local Ulam stability result which is introduced and proved in this paper. This result characterizes partially defined almost homomorphisms between two analytic compact groups with simple Lie algebras.

math.GR↗

arXiv2024

Asymptotic distribution for pairs of linear and quadratic forms at integral vectors

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 $n \ge 5$ then under the assumptions that for every $(α, β) \in \mathbb R^2 \setminus \{ (0,0) \}$, the form $αq + β\mathbf l^2$ is irrational and that the signature of the restriction of $q$ to the kernel of $\mathbf l$ is $(p, n-1-p)$, where $3\le p \le n-2$, the number of vectors $v \in \mathbb Z^n$ for which $\|v\| < T$, $a < q(v) < b$ and $c< \mathbf l(v) < d$ is asymptotically $$ C(q, \mathbf l)(d-c)(b-a)T^{n-3} , $$ as $T \to \infty$, where $C(q, \mathbf l)$ only depends on $q$ and $\mathbf l$. The density of the set of joint values of $(q, \mathbf l)$ under the same assumptions is shown by Gorodnik (2004).

math.DS↗

arXiv2023

Polynomiality of the faithful dimension of nilpotent groups over finite truncated valuation rings

The faithful dimension of a finite group $\mathrm G$ over $\mathbb C$, denoted by $m_\mathrm{faithful}(\mathrm G)$, is the smallest integer $n$ such that $\mathrm G$ can be embedded in $\mathrm{GL}_n(\mathbb C)$. Continuing our previous work (arXiv:1712.02019), we address the problem of determining the faithful dimension of a finite $p$-group of the form $\mathcal G_R:=\exp(\mathfrak g_R)$ associated to $\mathfrak g_R:=\mathfrak g \otimes_\mathbb Z R $ in the Lazard correspondence, where $\mathfrak g$ is a nilpotent $\mathbb Z$-Lie algebra and $R$ ranges over finite truncated valuation rings. Our first main result is that if $R$ is a finite field with $p^f$ elements and $p$ is sufficiently large, then $m_\mathrm{faithful}(\mathcal G_R)=fg(p^f)$ where $g(T)$ belongs to a finite list of polynomials $g_1,\ldots,g_k$, with non-negative integer coefficients. The list of polynomials is uniquely determined by the Lie algebra $\mathfrak g$. Furthermore, for $1\leq i\leq k$ the set of pairs $(p,f)$ for which $g=g_i$ is a finite union of Cartesian products $\mathcal P\times \mathcal F$, where $\mathcal P$ is a Frobenius set of prime numbers and $\mathcal F$ is a subset of $\mathbb N$ that belongs to the Boolean algebra generated by arithmetic progressions. Next we formulate a conjectural polynomiality property for $m_\mathrm{faithful}(\mathcal G_R)$ in the more general setting where $R$ is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras $\mathfrak g $ that are defined by partial orders, $m_\mathrm{faithful}(\mathcal G_R)$ is given by a single polynomial-type formula. Finally, we compute $m_\mathrm{faithful}(\mathcal G_R)$ precisely in the case where $\mathfrak g$ is the free metabelian nilpotent Lie algebra of class $c$ on $n$ generators and $R$ is a finite truncated valuation ring.

math.GR↗

arXiv2022

Compactifications of horospheric products

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 least three, which also leads to a precise description of the Busemann functions in terms of the points in the geometric compactification of each tree. We will discuss an application to the asymptotic behavior of integrable ergodic cocycles with values in the isometry group of such horospheric product.

math.GN↗

arXiv2020

Locally Random Groups

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 satisfying such a mixing inequality. In the same context, we will develop a Littlewood-Paley decomposition and explore its connection to the existence of the spectral gap for random walks. Moreover, under the dimension condition alone, we will prove a multi-scale entropy gain result `a la Bourgain-Gamburd and Tao.

math.GR↗

arXiv2019

Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups

Given a finite group $\mathrm{G}$ and a field $K$, the faithful dimension of $\mathrm{G}$ over $K$ is defined to be the smallest integer $n$ such that $\mathrm{G}$ embeds into $\mathrm{GL}_n(K)$. In this paper we address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_q:=\exp(\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q)$ associated to $\mathfrak{g}_q:=\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of $\mathscr{G}_p$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such on each part the faithful dimension of $\mathscr{G}_q$ for $q:=p^f$ is equal to $f g(p^f)$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory.

math.RT↗

arXiv2019

Some applications of arithmetic groups in cryptography

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.

math.GR↗

arXiv2018

Polynomial configurations in sets of positive upper density over local fields

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{R}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. We conclude that if $I \subseteq \mathbb{R}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ for an unbounded set of values $s \in \mathbb{R}$. It follows that the Borel chromatic number of the Cayley graph of $\mathbb{R}^m$ with respect to the set $\{ \pm F(s): s \in \mathbb{R} \}$ is infinite. Analogous results are also proven when $\mathbb{R}$ is replaced by the field of $p$-adic numbers $\mathbb{Q}_p$. At the end, we will also the existence of real analytic functions $f_1, \dots, f_m$, for which the analogous statements no longer hold.

math.CO↗

arXiv2018

On the chromatic number of structured Cayley graphs

In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, 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 the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings.

math.GR↗

arXiv2018

Future exchange rates and Siegel's paradox

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 a complete classification of all such aggregators in the general case. The formula obtained in this setting therefore describes all the negotiated no-arbitrage forward exchange rates in terms of a reciprocity function. Keywords: Siegel's paradox, forward exchange rates, discount bias.

q-fin.MF↗

arXiv2017

Positive harmonic functions of transformed random walks

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

math.PR↗

arXiv2017

Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points

We prove an analogue of a theorem of Eskin-Margulis-Mozes: suppose we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place and excluding the prime $2$, an irrational isotropic form ${\mathbf q}$ of rank $n\geq 4$ on $\mathbb{Q}_S$, a product of $p$-adic intervals $I_p$, and a product $Ω$ of star-shaped sets. We show that unless $n=4$ and ${\mathbf q}$ is split in at least one place, the number of $S$-integral vectors ${\mathbf v} \in {\mathsf{T}} Ω$ satisfying simultaneously ${\mathbf q}( {\mathbf v} ) \in I_p$ for $p \in S$ is asymptotically given by $$ λ({\mathbf q}, Ω) | I| \cdot \prod_{p\in S_f} T_p^{n-2},$$ as ${\mathsf{T}}$ goes to infinity, where $| I |$ is the product of Haar measures of the $p$-adic intervals $I_p$. The proof uses dynamics of unipotent flows on $S$-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an $S$-arithmetic variant of the $ α$-function introduced in the work of Eskin, Margulis, Mozes, and an $S$-arithemtic version of a theorem of Dani-Margulis.

math.DS↗