arXiv subjects

Menny Aka

Publications and source records attributed to Menny Aka.

At least 19 recordsLinked to original sources

arXiv2026

Planes in quadratic 4-space and associated shapes of lattices

Let $Q=-x_1^2-x_2^2-x_3^2+x_4^2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\mathbb{Q}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $\mathrm{SL}_2(\mathbb{Z}[i])\backslash \mathbb{H}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between $\mathrm{SO}_{1,3}(\mathbb{R})$ and $\mathrm{SL}_2(\mathbb{C})$ gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.

math.NT↗

arXiv2025

Fibonacci sequences in $\mathbb F_p$

We prove a conjecture due to Gica (Fibonacci Quarterly, 2008). Our proof is simple, uses only elementary abstract algebra, and generalizes to other recursive sequences.

math.NT↗

arXiv2024

Equidistribution of rational subspaces and their shapes

To any $k$-dimensional subspace of $\mathbb Q^n$ one can naturally associate a point in the Grassmannian ${\rm Gr}_{n,k}(\mathbb R)$ and two shapes of lattices of rank $k$ and $n-k$ respectively. These lattices originate by intersecting the $k$-dimensional subspace with the lattice $\mathbb Z^n$. Using unipotent dynamics we prove simultaneous equidistribution of all of these objects under a congruence conditions when $(k,n) \neq (2,4)$.

math.NT↗

arXiv2023

Seifert surfaces in the four-ball and composition of binary quadratic forms

We use composition of binary quadratic forms to systematically create pairs of Seifert surfaces that are non-isotopic in the four-ball. Our main topological result employs Gauss composition to classify the pairs of binary quadratic forms that arise as the Seifert forms of pairs of disjoint Seifert surfaces of genus one. The main ingredient of the proof is number-theoretic and of independent interest. It establishes a new connection between the Bhargava cube and the geometric approach to Gauss composition via planes in the space of two-by-two matrices. In particular, we obtain a geometric recipe that given any two binary quadratic forms finds a Bhargava cube that gives rise to their composition.

math.GT↗

arXiv2022

Joinings classification and applications [after Einsiedler and Lindenstruass]

This is the text accompanying my Bourbaki seminar on the work of Einsiedler and Lindenstrauss on joinings. The first five sections surveys their proof of the classification of joinings of higher-rank torus actions on arithmetic quotients of semisimple or perfect algebraic groups. The last section surveys how this classification can be used to tackle arithmetic problems, together with a new application for higher rank simple groups.

math.DS↗

arXiv2021

Simultaneous supersingular reductions of CM elliptic curves

We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show -- under additional congruence assumptions on the CM order -- that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut-Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.

math.NT↗

arXiv2021

Planes in four space and four associated CM points

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian Gr(2,4) and four lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement and the second two essentially arise from the accidental local isomorphism between SO(4) and SO(3)xSO(3). As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings we prove simultaneous equidistribution of all of these objects under two splitting conditions.

math.NT↗

arXiv2019

Continued Fractions of Arithmetic Sequences of Quadratics

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach the normal statistics given by the Gauss-Kuzmin measure. Under the generalized Riemann hypothesis, we prove that there exist full density subsets S of P and T of N satisfying the same assertion. We give a rate of convergence in all cases.

math.NT↗

arXiv2019

On effective equidistribution for quotients of SL(d)

We prove the first case of polynomially effective equidistribution of closed orbits of semisimple groups with nontrivial centralizer. The proof relies on uniform spectral gap, builds on, and extends work of Einsiedler, Margulis, and Venkatesh.

math.DS↗

arXiv2017

Diophantine approximation on matrices and Lie groups

We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine exponent of random subgroups of certain nilpotent Lie groups in terms of representation theoretic data.

math.NT↗

arXiv2015

Integer points on spheres and their orthogonal grids

The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics we obtained the desired equidistribution result in dimension d>5 and in dimension d=4,5 under a mild congruence condition on the square of the radius. The case of d=3 is considered in a separate paper.

math.NT↗

arXiv2015

Integer points and their orthogonal lattices

Linnik proved in the late 1950's the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition.

math.NT↗

arXiv2015

On metric diophantine approximation in matrices and Lie groups

We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over the rationals. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.

math.NT↗

arXiv2014

Diophantine properties of nilpotent Lie groups

A finitely generated subgroup Γ of a real Lie group G is said to be Diophantine if there is β> 0 such that non-trivial elements in the word ball B_Γ(n) centered at the identity never approach the identity of G closer than |B_Γ (n)|^{-β}. A Lie group G is said to be Diophantine if for every k > 0, a random k-tuple in G generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non Diophantine nilpotent and solvable (non nilpotent) Lie groups.

math.GR↗

arXiv2014

Duke's Theorem for subcollections

We combine effective mixing and Duke's Theorem on closed geodesics on the modular surface to show that certain subcollections of the collection of geodesics with a given discriminant still equidistribute. These subcollections are only assumed to have sufficiently large total length without any further restrictions.

math.DS↗

arXiv2013

On the evolution of continued fractions in a fixed quadratic field

We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the `normal' statistics given by the Gauss-Kuzmin measure. As a by-product, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer $k$ and a quadratic irrational $\al$, the length of the period of the continued fraction expansion of $k^n\al$ equals $c k^n +o(k^{(1-\frac{1}{16})n})$ for some positive constant $c$. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on $S$-arithmetic homogeneous spaces.

math.DS↗

arXiv2012

Homogeneous number of free generators

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group Γwhich admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group Γis uncountable.

math.GR↗

arXiv2011

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the the fibers are of unbounded size.

math.GR↗