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Shrey Sanadhya

Publications and source records attributed to Shrey Sanadhya.

11 recordsLinked to original sources
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↗
arXiv2026

Subdiagrams and invariant measures for generalized Bratteli diagrams

The results of this paper contribute to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior.

math.DS↗
arXiv2024

Periodicity of joint co-tiles in $\mathbb{Z}^d$

An old theorem of Newman asserts that any tiling of $\mathbb{Z}$ by a finite set is periodic. A few years ago, Bhattacharya proved the periodic tiling conjecture in $\mathbb{Z}^2$. Namely, he proved that for a finite subset $F$ of $\mathbb{Z}^2$, if there exists $A \subseteq \mathbb{Z}^2$ such that $F \oplus A = \mathbb{Z}^2$ then there exists a periodic $A' \subseteq \mathbb{Z}^2$ such that $F \oplus A' = \mathbb{Z}^2$. The recent refutation of the periodic tiling conjecture in high dimensions due to Greenfeld and Tao motivates finding different generalizations of Newman's theorem and of Bhattacharya's theorem that hold in arbitrary dimension $d$. In this paper, we formulate and prove such generalizations. We do so by studying the structure of joint co-tiles in $\mathbb{Z}^d$. Our generalization of Newman's theorem states that for any $d \ge 1$, any joint co-tile for $d$ independent tiles is periodic. For a $(d-1)$-tuple of finite subsets of $\mathbb{Z}^d$ that satisfy a certain technical condition that we call property $(\star)$, we prove that any joint co-tile decomposes into disjoint $(d-1)$-periodic sets. Consequently, we show that for a $(d-1)$-tuple of finite subsets of $\mathbb{Z}^d$ that satisfy property $(\star)$, the existence of a joint co-tile implies the existence of periodic joint co-tile. Conversely, we prove that if a finite subset $F$ in $\mathbb{Z}^d$ admits a periodic co-tile $A$, then there exist $(d-1)$ additional tiles that together with $F$ are independent and admit $A$ as a joint co-tile, so that the first $(d-2)$ of these tiles together with $F$ satisfy property $(\star)$. Combined, our results give a new necessary and sufficient condition for a subset of $\mathbb{Z}^d$ to tile periodically. We also discuss tilings and joint tilings in other countable abelian groups.

math.DS↗
arXiv2024

Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the $2$-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.

math.DS↗
arXiv2024

Substitution-dynamics and invariant measures for infinite alphabet-path space

We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.

math.DS↗
arXiv2023

Bratteli diagrams in Borel dynamics

Bratteli-Vershik models have been very successfully applied to the study of various dynamical systems, in particular, in Cantor dynamics. In this paper, we study dynamics on the path spaces of generalized Bratteli diagrams that form models for non-compact Borel dynamical systems. Generalized Bratteli diagrams have countably infinite many vertices at each level, thus the corresponding incidence matrices are also countably infinite. We emphasize differences (and similarities) between generalized and classical Bratteli diagrams. Our main results: $(i)$ We utilize Perron-Frobenius theory for countably infinite matrices to establish criteria for the existence and uniqueness of tail-invariant path space measures (both probability and $σ$-finite). $(ii)$ We provide criteria for the topological transitivity of the tail equivalence relation. $(iii)$ We describe classes of stationary generalized Bratteli diagrams (hence Borel dynamical systems) that: $(a)$ do not support a probability tail-invariant measure, $(b)$ are not uniquely ergodic with respect to the tail equivalence relation. $(iv)$ We describe classes of generalized Bratteli diagrams which can or cannot admit a continuous Vershik map and construct a Vershik map which is a minimal homeomorphism of a (non locally compact) Polish space. $(v)$ We provide an application of the theory of stochastic matrices to analyze diagrams with positive recurrent incidence matrices.

math.DS↗
arXiv2022

A note on reduction of tiling problems

We show that translational tiling problems in a quotient of $\mathbb{Z}^d$ can be effectively reduced or ``simulated'' by translational tiling problems in $\mathbb{Z}^d$. In particular, for any $d \in \mathbb{N}$, $k < d$ and $N_1,\ldots,N_k \in \mathbb{N}$ the existence of an aperiodic tile in $\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$ implies the existence of an aperiodic tile in $\mathbb{Z}^d$. Greenfeld and Tao have recently disproved the well-known periodic tiling conjecture in $\mathbb{Z}^d$ for sufficiently large $d \in \mathbb{N}$ by constructing an aperiodic tile in $\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$ for suitable $d,N_1,\ldots,N_k \in \mathbb{N}$.

math.CO↗
arXiv2022

A shrinking target theorem for ergodic transformations of the unit interval

We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\overset{\infty}{\cap}} \, \underset{i=n}{\overset{\infty}{\cup}}\, f^{-i}(B (R_α^{i} x,b_i))$$ has full Lebesgue measure for almost every $x \in [0,1)$ and almost every $α\in [0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at $x \in [0,1)$ and $R_α: [0,1) \rightarrow [0,1)$ is rotation by $α\in [0,1)$. As a corollary, we provide partial answer to a question asked by Chaika in the context of interval exchange transformations.

math.DS↗
arXiv2021

Cohomology of hyperfinite Borel actions

We study cocycles of countable groups $Γ$ of Borel automorphisms of a standard Borel space $(X, \mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $Γ$ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the $2$-odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup $H$ of $G$. We also provide a Borel version of Gottschalk-Hedlund theorem.

math.DS↗
arXiv2021

Coboundaries of commuting Borel automorphisms

We show that if $S,T$ are two commuting automorphisms of standard Borel space such that they generate a free Borel $\Z^2$-action then $S$ and $T$ do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel $\Z^d$-actions.

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