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Simon Machado

Publications and source records attributed to Simon Machado.

18 recordsLinked to original sources
arXiv2026

Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\"{u}ck-Zimmer Theorem

Fraczyk and Gelander proved in \cite{FG} that for any simple Lie group $G$ of high rank and for every non-lattice discrete subgroup $\Gamma\leq G$, the injectivity radius of points in $G/\Gamma$ is unbounded, resolving a conjecture of Margulis. In this work we obtain an explicit lower bound on the growth rate of the maximal injectivity radius of points taken from growing balls in $G/\Gamma$. More explicitly, we prove that for any $R>0$, one can embed a ball of radius $c\log^{(4)}R$ in $G/\Gamma$ centered at some point $[g]\in G/\Gamma$ where $g$ is taken from $G_R$ and for some constant $c=c(G,\Gamma)$. In particular, we show that for a general discrete subgroup $\Gamma$, if the injectivity radius growth in $G/\Gamma$ is slower than $\log^{(4)}$, $\Gamma$ must be a lattice. Additionally, we give a new, shorter and simpler proof of the Nevo-St\"{u}ck-Zimmer Theorem, saying that every action of a high rank simple group with property $(T)$ is either essentially free or essentially transitive. The results in this paper are obtained using the almost structure of measures from the accompanying paper, together with additional geometric considerations. As a step in the proof, we develop the following characterization for lattices. A discrete subgroup $\Gamma\leq G$ is a lattice if and only if there is a probability measure on $G/\Gamma$ which is sufficiently almost invariant under $G$. More precisely, suppose $\Gamma\leq G$ is a discrete subgroup for which there exists a probability measure $\nu$ on $G/\Gamma$ for which $W_1^{b}(g\nu,\nu)\leq \eps_0$ for some $\eps_0(\Gamma)>0$, then $\Gamma$ is a lattice.

math.DS↗
arXiv2026

The Structure of Almost Stationary Measures

Let $G$ be a higher-rank simple Lie group acting on a space $X$. A theorem of Nevo and Zimmer asserts that every ergodic stationary probability measure on $X$ is either $G$-invariant or admits a projective factor $G/Q$ for a proper parabolic subgroup $Q$. We develop a quantitative theory of stationary measures and prove an effective form of this dichotomy. We introduce notions of $\eps$-almost stationarity, $\delta$-almost invariance and $\delta$-almost projective factor, and show that every $\eps$-almost stationary measure is either $\delta$-almost invariant or carries a $\delta'$-almost projective factor, with $\delta,\delta'$ explicit in $\eps$ and depending only on $G$. No ergodicity, arithmeticity or Diophantine hypothesis is imposed, and the bounds are uniform over all $G$-spaces. The proof introduces several tools: the \emph{entropigeonhole method}, an entropy-based pigeonhole principle yielding a quantitative Mautner phenomenon; \emph{factor functions}, quantitative analogues of functions on homogeneous factor spaces; and a \emph{fast generation} dichotomy in the spirit of growth in groups. In a companion paper these are used to show, among other things, that a discrete subgroup of infinite covolume has injectivity radius at least $c\log^{(4)}r$ somewhere in the ball of radius $r$, which is an effective form of a theorem of Fr\k{a}czyk and Gelander.

math.DS↗
arXiv2026

${\mathrm{ASL}_n}(\mathbb Z)$ invariant random subsets of $\mathbb Z^n$

We classify measures on $\{0,1\}^{\mathbb{Z}^d}$, $d \geq 3$, the space of subsets of $\mathbb{Z}^d$, which are invariant under all affine special linear transformations. In other words, we classify simple point processes on $\mathbb{Z}^d$ whose law is invariant under affine special linear transformations. We show that every such process is built from a random equivariant polynomial together with independent random sampling, a higher-order generalisation of the cut-and-project method: a random polynomial map is drawn from a distribution invariant under a natural action of $\mathrm{SL}_d(\mathbb{Z})$, each site is then retained independently with a probability determined by a measurable function of the polynomial's value, and the classical cut-and-project construction is recovered in the degree-one case. As a corollary, when the underlying $\mathbb{Z}^d$-action is weakly mixing the measure must be a convex combination of Bernoulli shifts, in the spirit of de Finetti's theorem on exchangeable processes. Our theorem also makes precise how the Howe--Moore theorem fails for the pair $(\mathrm{ASL}_d(\mathbb{Z}), \mathrm{SL}_d(\mathbb{Z}))$. Motivated by this classification, we formulate a conjecture for $\mathrm{ASL}_d(\mathbb{R})$-invariant point processes on $\mathbb{R}^d$, predicting that any such set decomposes into a Poisson part and a quasicrystal part. The proofs rely on the interaction between the Host--Kra theory of characteristic factors, Zimmer's theory of dynamical cocycles of simple Lie groups, and the dynamics of $\mathrm{SL}_d(\mathbb{Z})$-actions on homogeneous spaces.

math.PR↗
arXiv2026

On the structure of approximate rings

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate subrings. Our aim is to develop a general framework for the sum-product phenomenon that applies uniformly across arbitrary rings. The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication. Another application of the main structure theorem is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24]. This existence theorem extends beyond the finite (and pseudofinite) setting. To illustrate the scope of the method, we also establish a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing a classical sum-product result of Meyer.

math.RA↗
arXiv2026

Quasihomomorphisms to real algebraic groups

A quasihomomorphism is a map that satisfies the homomorphism relation up to bounded error. Fujiwara and Kapovich proved a rigidity result for quasihomomorphisms taking values in discrete groups, showing that all quasihomomorphisms can be built from homomorphisms and sections of bounded central extensions. We study quasihomomorphisms with values in real linear algebraic groups, and prove an analogous rigidity theorem.

math.GR↗
arXiv2026

The non-ergodic Host-Kra-Ziegler structure theorem for $\mathbb{Z}^d$-actions via measurable selections

We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\ge 2$ due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-Ziegler structure theorem is a key input in the companion paper by the second author and Fraczyk classifying point processes (i.e. random subsets) of $\mathbb{Z}^d$ whose law is invariant under the group $\mathrm{ASL}_d(\mathbb{Z})$ of affine transformations.

math.DS↗
arXiv2025

Asymmetric stability of the Brunn--Minkowski inequality in compact Lie groups

We show a stability result for the recently established Brunn--Minkowski inequality in compact simple Lie groups. Namely, we prove that if two compact subsets $A, B$ of a compact simple Lie group $G$ satisfy $$ \mu(AB)^{1/d'} \leq (1 + \epsilon)\left(\mu(A)^{1/d'} + \mu(B)^{1/d'}\right)$$ where $AB$ is the Minkowski product $\{ab : a \in A, b \in B\}$, $d'$ denotes the minimal codimension of a proper closed subgroup and $\mu$ is a Haar measure, then $A$ and $B$ must approximately look like neighbourhoods of a proper subgroup $H$ of codimension $d'$, with an error that depends quantitatively on $d', \epsilon$ and the ratio $\frac{\mu(A)}{\mu(B)}$. This result implies an improved error rate in the Brunn--Minkowski inequality in compact simple Lie groups $$\mu(AB)^{\frac{1}{d'}} \geq (1-C\mu(A)^{\frac{2}{d'}})\left(\mu(A)^{\frac{1}{d'}} + \mu(B)^{\frac{1}{d'}}\right) $$ sharp, up to the constant $C$ which depends on $d'$ and $\frac{\mu(A)}{\mu(B)}$ alone. Our approach builds upon an earlier paper of the author proving the Brunn--Minkowski inequality, and stability in the case $A=B$. We employ a combinatorial multi-scale analysis and study so-called density functions. Additionally, the asymmetry between $A$ and $B$ introduces new challenges, requiring the use of non-abelian Fourier theory and stability results for the Pr\'ekopa--Leindler inequality.

math.GR↗
arXiv2024

Language Models as Science Tutors

NLP has recently made exciting progress toward training language models (LMs) with strong scientific problem-solving skills. However, model development has not focused on real-life use-cases of LMs for science, including applications in education that require processing long scientific documents. To address this, we introduce TutorEval and TutorChat. TutorEval is a diverse question-answering benchmark consisting of questions about long chapters from STEM textbooks, written by experts. TutorEval helps measure real-life usability of LMs as scientific assistants, and it is the first benchmark combining long contexts, free-form generation, and multi-disciplinary scientific knowledge. Moreover, we show that fine-tuning base models with existing dialogue datasets leads to poor performance on TutorEval. Therefore, we create TutorChat, a dataset of 80,000 long synthetic dialogues about textbooks. We use TutorChat to fine-tune Llemma models with 7B and 34B parameters. These LM tutors specialized in math have a 32K-token context window, and they excel at TutorEval while performing strongly on GSM8K and MATH. Our datasets build on open-source materials, and we release our models, data, and evaluations.

cs.CL↗
arXiv2024

Minimal doubling for small subsets in compact Lie groups

We prove a sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group. Namely, let $G$ be a compact connected Lie group of dimension $d_G$, we show that for for all measurable subsets $A$, we have $$\mu_G(A^2) \geq \left(2^{d_G-d_H} - C\mu_G(A)^{\frac{2}{d_G-d_H}}\right)\mu_G(A)$$ where $d_H$ is the maximal dimension of a proper closed subgroup $H$ and $C > 0$ is a dimensional constant. This settles a conjecture of Breuillard and Green, and recovers and improves - with completely different methods - a recent result of Jing--Tran--Zhang corresponding to the case $G=SO_3(\mathbb{R})$. As is often the case, the above doubling inequality stems from a special case of general product-set estimates. We prove that for all $\epsilon >0$ and for any pair of sufficiently small measurable subsets $A,B$ a Brunn--Minkowski-type inequality holds: $$ \mu_G(AB)^{\frac{1}{d_G-d_H}} \geq (1-\epsilon)\left( \mu_G(A)^{\frac{1}{d_G-d_H}} + \mu_G(B)^{\frac{1}{d_G-d_H}}\right).$$ Going beyond the scope of the Breuillard--Green conjecture, we prove a stability result asserting that the only subsets with close to minimal doubling are essentially neighbourhoods of proper subgroups i.e. of the form $$H_{\delta}:=\{g \in G: d(g,H)<\delta\}$$ where $H$ denotes a proper closed subgroup of maximal dimension, $d$ denotes a bi-invariant distance on $G$ and $\delta >0$. Our approach relies on a combination of two toolsets: optimal transports and its recent applications to the Brunn--Minkowski inequality, and the structure theory of compact approximate subgroups.

math.GR↗
arXiv2023

Approximate lattices and S-adic linear groups

We provide and motivate in this paper a natural framework for the study of approximate lattices. Namely, we consider approximate lattices in so-called $S$-adic linear groups and define relevant notions of arithmeticity. We also adapt to this framework classical results of the theory of lattices and Meyer sets. Results from this paper will play a role in the proof of a structure theorem for approximate lattices in $S$-adic linear groups which is the subject of a companion paper. We extend a theorem of Schreiber's concerning the coarse structure of approximate subgroups in Euclidean spaces to approximate subgroups of unipotent $S$-adic groups. We generalise Meyer's structure theorem for approximate lattices in locally compact abelian groups to a precise structure theorem for approximate lattices in unipotent $S$-adic groups. Finally, we study intersections of approximate lattices of $S$-adic linear groups with certain subgroups such as the nilpotent radical and Levi subgroups. We furthermore show that the framework of $S$-adic linear groups enables us to provide statements more precise than earlier results.

math.NT↗
arXiv2023

The definitions of approximate lattices

Approximate lattices of locally compact groups were first studied in a seminal monograph of Yves Meyer and were subsequently used in the theory of aperiodic order to model objects such as Pisot numbers, quasi-cristals or aperiodic tilings. Meyer studied approximate lattices of Euclidean spaces - now dubbed Meyer sets. A fascinating feature of this theory is the wealth of natural and equivalent definitions of Meyer sets that are available. The study of approximate lattices in non-commutative groups has recently attracted more attention. They are meant as generalisation of lattices of locally compact groups (i.e. discrete subgroup of finite co-volume) and, as such, are roughly defined as approximate subgroups of locally compact groups that are both discrete and have finite co-volume. In this generalized framework, the correct notion of `finite co-volume' is however up for interpretation. A number of definitions recently arose following work of Bj\"{o}rklund and Hartnick, Hrushovski, and the author. In stark contrast with the Euclidean framework, little is known of the relations between these notions. Our main objective in this paper is to address this issue. We establish clear relations between these definitions and show that, in fact, these definitions are not all equivalent. We prove nevertheless that all the notions mentioned are related to one another and simply represent varying degrees of generality. Among the original results we prove here, we extend a cornerstone theorem of Meyer and show that all strong approximate lattices are laminar; we build measures with weak invariance properties on the invariant hull of approximate lattices; we use quasi-models to show that any BH-approximate lattice is contained in an approximate lattice; we prove that even laminar approximate lattices can be arbitrarily far from strong approximate lattices and model sets.

math.GR↗
arXiv2023

Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces

In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus g. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in [Jiang-Tidor-Yao-Zhang-Zhao, 2021] for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the ``approximate multiplicity'' of eigenvalues, i.e., the number of eigenvalues in windows of size $1/\log^\beta(g)$, $\beta>0$. This work provides new insights on a conjecture by Colin de Verdi{\`e}re [Colin de Verdi{\`e}re, 1986] and new ways to transfer spectral results from graphs to surfaces.

math.SP↗
arXiv2023

The structure of approximate lattices in linear groups

Approximate lattices are aperiodic generalisations of lattices of locally compact groups that were first studied in seminal work of Yves Meyer. They are defined as those uniformly discrete approximate subgroups (symmetric subsets stable under multiplication up to a finite error) of locally compact groups that have finite co-volume. Meyer showed that approximate lattices of Euclidean spaces (a.k.a. Meyer sets) are related to lattices in higher-dimensional Euclidean spaces via the cut-and-project construction. A fundamental challenge of the theory of approximate lattices is to extend Meyer's theorem beyond Euclidean spaces. Our main result provides a complete structure theorem for approximate lattices valid in all linear algebraic groups over local fields and their finite products, in particular providing the most general extension of Meyer's theorem to date. Our proof relies on an extension of a theorem of Lubotzky--Mozes--Raghunathan to approximate lattices in S-adic semi-simple groups, a notion of cohomology tailored to the study of approximate subgroups, a universality statement complementing a recent result of Hrushovski and a generalisation of a result of Burger and Monod about bounded cohomology of lattices.

math.GR↗
arXiv2023

Small doubling for discrete subsets of non-commutative groups and a theorem of Lagarias

Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset $X$ of $\mathbb{R}^n$ is relatively dense and $X - X$ is uniformly discrete, then $X$ is an approximate lattice. We prove two generalisation of Lagarias' theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.

math.GR↗
arXiv2020

Closed approximate subgroups: compactness, amenability and approximate lattices

We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices i.e. those approximate subgroups that are discrete and have finite co-volume. We prove an approximate subgroup version of Cartan's closed-subgroup theorem and study some applications. We give a structure theorem for closed approximate subgroups of amenable groups in the spirit of the Breuillard--Green--Tao theorem. We then prove two results concerning approximate lattices: we extend to amenable groups a structure theorem for mathematical quasi-crystals due to Meyer; we prove results concerning intersections of radicals of Lie groups and discrete approximate subgroups generalising theorems due to Auslander, Bieberbach and Mostow. As an underlying theme, we exploit the notion of good models of approximate subgroups that stems from the work of Hrushovski, and Breuillard, Green and Tao. We show how one can draw information about a given approximate subgroup from a good model, when it exists.

math.GR↗
arXiv2019

Infinite Approximate Subgroups of Soluble Lie Groups

We study infinite approximate subgroups of soluble Lie groups. Generalising a theorem of Fried and Goldman we show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building up on this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer.

math.GR↗
arXiv2018

Approximate Lattices and Meyer Sets in Nilpotent Lie Groups

We show that uniform approximate lattices in nilpotent Lie groups are subsets of model sets. This extends a theorem due to Yves Meyer about quasicrystals in Euclidean spaces. To do so we study relatively dense subsets of simply connected nilpotent Lie groups and their logarithms. We then deduce a simple criterion for the existence of an approximate lattice in a given nilpotent Lie group.

math.GR↗