arXiv subjects

Jonathan Pakianathan

Publications and source records attributed to Jonathan Pakianathan.

At least 19 recordsLinked to original sources

arXiv2026

Configuration Spaces and Braid Groups

The main thrust of these notes is 3-fold: (1) An analysis of certain $K(\pi,1)$'s that arise from the connections between configuration spaces, braid groups, and mapping class groups, (2) a function space interpretation of these results, and (3) a homological analysis of the cohomology of some of these groups for genus zero, one, and two surfaces possibly with marked points, as well as the cohomology of certain associated function spaces. An example of the type of results given here is an analysis of the space k particles moving on a punctured torus up to equivalence by the natural $SL(2,\mathbb{Z})$ action.

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arXiv2024

A quantitative version of the Steinhaus theorem

The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if $A \subset {\Bbb R}^d$ has positive Lebesgue measure than $A-A=\{x-y: x,y \in A\}$ contains an open ball. We obtain some quantitative lower bounds on the size of this ball and in some cases, relate it to natural geometric properties of $\partial A$. We also study the process $K_n =\frac{1}{2}(K_{n-1} - K_{n-1})$ when $K_0$ is a compact subset of $\mathbb{R}^d$ and determine various aspects of its convergence to $Conv(K_1)$, the convex hull of $K_1$. We discuss some connections with convex geometry, Weyl tube formula and the Kakeya needle problem. \noindent {\it Keywords: Measure theory, Steinhaus theorem, Convex geometry, Weyl tube formula.} \noindent 2020 {\it Mathematics Subject Classification:} Primary: 28A75, 52A27. Secondary: 52A30, 53A07.

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arXiv2024

Random walks and the "Euclidean" association scheme in finite vector spaces

In this paper, we provide an application to the random distance-$t$ walk in finite planes and derive asymptotic formulas (as $q \to \infty$) for the probability of return to start point after $\ell$ steps based on the "vertical" equidistribution of Kloosterman sums established by N. Katz. The application of these deep results from number theory allow a determination of the second order terms in the answers that simpler spectral gap/mixing rate methods do not. This work relies on a "Euclidean" association scheme studied in prior work of W.M.Kwok, E. Bannai, O. Shimabukuro and H. Tanaka. We also provide a self-contained discussion of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

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arXiv2019

Random gap processes and asymptotically complete sequences

We study a process of generating random positive integer weight sequences $\{ W_n \}$ where the gaps between the weights $\{ X_n = W_n - W_{n-1} \}$ are i.i.d. positive integer-valued random variables. We show that as long as the gap distribution has finite $\frac{1}{2}$-moment, almost surely, the resulting weight sequence is asymptotically complete, i.e., all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights. We then show a much stronger result that if the gap distribution has a moment generating function with large enough radius of convergence, then every large enough multiple of the gcd of gap values can be written as a sum of $m$ distinct weights for any fixed $m \geq 2$.

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arXiv2018

Cayley Digraphs Associated to Arithmetic Groups

We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\mathbb{F}_q), d \geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix, and on whether $q$ is congruent to $1$ or $3$ (mod $4$), but independent of $q$ otherwise.

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arXiv2017

A variant of Waring's Problem for the ring of integers modulo n

We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in $\mathbb{Z}_n$ can be written as a sum of three squares. Hensel's Theorem for $p$-adic integers plays a key role. Finally, we give an application of this problem to the Erd\H os-Falconer distance problem for rings $\mathbb{Z}_n^d$.

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arXiv2017

Direction sets, Lipschitz graphs and density

We consider the direction set determined by various subsets $E$ of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph of a Lipschitz function and the direction set is not dense in the sphere, (ii) The subset is the graph of a non-Lipschitz function and the direction set is dense but not everything, or (iii) The subset is not a graph (in a suitable sense) and every direction is determined by the set. We then explore a variety of results based on this trichotomy under additional assumptions on the set $E$.

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arXiv2015

On directions determined by subsets of vector spaces over finite fields

We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a $k$-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a $k$-dimensional hyperplane shows. We can view this question as an Erd\H os type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. For discrete subsets of ${\Bbb R}^d$, this question has been previously studied by Pach, Pinchasi and Sharir.

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arXiv2015

The Fuglede Conjecture holds in ${\Bbb Z}_p \times {\Bbb Z}_p$

In this paper we study subsets $E$ of ${\Bbb Z}_p^d$ such that any function $f: E \to {\Bbb C}$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in ${\Bbb Z}_p^2$ which says that $E \subset {\Bbb Z}_p^2$ is spectral if and only if $E$ tiles ${\Bbb Z}_p^2$ by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.

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arXiv2014

On the distribution of distances in homogeneous compact metric spaces

We provide a simple proof that in any homogeneous, compact metric space of diameter $D$, if one finds the average distance $A$ achieved in $X$ with respect to some isometry invariant Borel probability measure, then $$\frac{D}{2} \leq A \leq D.$$ This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds. We then classify the cases where one of the extremes occurs. In particular any homogeneous compact metric space where $A=\frac{D}{2}$ possesses a strict antipodal property which implies in particular that the distribution of distances in $X$ is symmetric about $\frac{D}{2}$ which is hence both mean and median of the distribution. In particular, we show that the only closed, connected, positive-dimensional Riemannian manifolds with this strict antipodal property are spheres.

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arXiv2014

A note on the unit distance problem for planar configurations with Q-independent direction set

Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must be $\mathbb{Q}$-independent. This is related to the Erdos unit distance problem but with a simplifying additional assumption on the direction set which holds "generically". We show that $T(n+1)-T(n)$ is the Hamming weight of $n$, i.e., the number of nonzero binary coefficients in the binary expansion of $n$, and find a formula for $T(n)$ explicitly. In particular $T(n)$ is $Θ(n log(n))$. Furthermore we describe a process to construct a set of $n$ points in the plane with $\mathbb{Q}$-independent unit length direction set which achieves exactly $T(n)$ unit distances. In the process of doing this, we show $T(n)$ is also the same as the maximum number of edges a subset of vertices of size $n$ determines in either the countably infinite lattice $\mathbb{Z}^{\infty}$ or the infinite hypercube graph $\{0,1\}^{\infty}$. The problem of determining T(n) can be viewed as either a type of packing or isoperimetric problem.

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arXiv2013

On a canonical construction of tesselated surfaces via finite group theory, Part I

This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth surfaces equipped with a closed-cell structure which is face and edge transitive and which has at most 2 orbits of vertices. These tesselated surfaces can also be viewed as abstract 3-polytopes (or as graph embeddings in the corresponding surface) which are either equivar or dual to abstract quasiregular polytopes. This construction generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to Σ_6 it yields 4477 tesselated surfaces of 27 distinct genus and even more varieties of tesselation cell structure. We study the distribution of these surfaces in various groups and some interesting resulting tesselations. In a second paper, we show that extensions of groups result in branched coverings between the component surfaces in their decompositions. We also exploit functoriality to obtain interesting faithful, orientation preserving actions of subquotients of these groups and their automorphism groups on these surfaces and in the corresponding mapping class groups.

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arXiv2013

On a canonical construction of tesselated surfaces via finite group theory, Part II

This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to Σ_6 it yields 4477 tesselated surfaces of 27 distinct genus and even more varieties of tesselation cell structure. These tesselations are face and edge transitive and consist of closed cell structures. In this paper, we continue to study the distribution of these surfaces in various groups and some interesting resulting tesselations with the aid of computer computations. We also show that extensions of groups result in branched coverings between the component surfaces in their decompositions. Finally we exploit functoriality to obtain interesting faithful, orientation preserving actions of subquotients of these groups and their automorphism groups on these surfaces and in the corresponding mapping class groups.

math.GT↗

arXiv2013

Lie algebras and Higher torsion in p-groups

We study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence E_r^{*,*}[g] is defined for any Lie algebra g which models the Bockstein spectral sequence of the corresponding group in characteristic p. This spectral sequence is then studied for complex semisimple Lie algebras like sl_n(C), and the results there are transferred to the corresponding p-group via the intermediary arithmetic Lie algebra defined over Z. Over C, it is shown that E_1^{*,*}[g]=H^*(g,U(g)^*)=H^*(ΛBG) where U(g)^* is the dual of the universal enveloping algebra of g and ΛBG is the free loop space of the classifying space of a Lie group G associated to g. In characteristic p, a phase transition is observed. For example, it is shown that the algebra E_1^{*,*}[sl_2[F_p]] requires at least 17 generators unlike its characteristic zero counterpart which only requires two.

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arXiv2013

Kakeya Configurations in Lie Groups and Homogeneous Spaces

In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line (respectively an interval of unit length) through all directions {\bf continuously, without repeating a "direction"}.) We show under some general assumptions that any such continuous Kakeya line configuration set in a connected Lie group must contain an open neighborhood of the identity, and hence must have positive Haar measure. In connected nilpotent Lie groups $G$, the only subspace of $G$ that contains such an unoriented line configuration is shown to be $G$ itself. Finally some similar questions in homogeneous spaces are addressed. These questions were motivated by work of Z. Dvir in the finite field setting.

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arXiv2012

Bockstein Closed 2-Group Extensions and Cohomology of Quadratic Maps

A central extension of the form $E: 0 \to V \to G \to W \to 0$, where $V$ and $W$ are elementary abelian 2-groups, is called Bockstein closed if the components $q_i \in H^*(W, \FF_2)$ of the extension class of $E$ generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of $G$ when $E$ is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of $G$ has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg-Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps $Q : W \to V$ associated to the extensions $E$ of the above form.

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arXiv2012

Three-point configurations determined by subsets of $\mathbb{F}_q^2$ via the Elekes-Sharir paradigm

We prove that if $E \subset {\mathbb F}_q^2$, $q \equiv 3 \mod 4$, has size greater than $Cq^{7/4}$, then $E$ determines a positive proportion of all congruence classes of triangles in ${\mathbb F}_q^2$. The approach in this paper is based on the approach to the Erd\H os distance problem in the plane due to Elekes and Sharir, followed by an incidence bound for points and lines in ${\mathbb F}_q^3$. We also establish a weak lower bound for a related problem in the sense that any subset $E$ of ${\mathbb F}_q^2$ of size less than $cq^{4/3}$ definitely does not contain a positive proportion of {\bf translation} classes of triangles in the plane. This result is a special case of a result established for $n$-simplices in ${\mathbb F}_q^d$. Finally, a necessary and sufficient condition on the lengths of a triangle for it to exist in $\mathbb{F}^2$ for any field $\mathbb F$ of characteristic not equal to 2 is established as a special case of a result for $d$-simplices in ${\mathbb F}^d$.

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arXiv2011

Exponents of Zero divisors in the Cohomology ring of a finite group

It is well known that the positive degree cohomology of a finite group G is annihilated by |G|. We improve on this bound in the case of odd degree elements in the integer cohomology ring and show that $e_{odd}(G)$, the exponent of the $\oplus_{k=0}^{\infty} H^{2k+1}(G,\mathbb{Z})$ satisfies $e_{odd}(G)^2$ divides 2|G| and in particular $e_{odd}(G) \leq \sqrt{2|G|}.$ We also provide examples to show this bound for $e_{odd}(G)$ is sharp as a general bound over all finite groups G. The result comes from a fact about zero divisors having "complementary exponent" which we prove using duality in Tate cohomology. More particularly if $α, β$ are elements of positive degree in $H^*(G,\mathbb{Z})$ satisfying $αβ= 0$ then the order of $β$, $o(β)$ divides $\frac{|G|}{o(α)}$. We also apply this fact to get some results on elements of exceptionally high exponent in the cohomology ring.

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