arXiv subjects

Felipe A. Ramirez

Publications and source records attributed to Felipe A. Ramirez.

15 recordsLinked to original sources

arXiv2026

Duffin--Schaeffer examples, real residue systems, and Bohr-set primes

We prove the following generalization of a well-known result of Duffin and Schaeffer: For any given countable sets $Y \subset\mathbb{R}$ and $Z\subset\mathbb{R}\setminus\operatorname{span}_\mathbb{Q}(\{1\}\cup Y)$, there exist functions $\psi$ such that the set of inhomogeneously $\psi$-approximable numbers has zero measure or full measure, according as the inhomogeneous parameter lies in $Y$ or $Z$. The proof uses an analogue of residue systems where the residues can take arbitrary real values, and it also requires information about the distribution of primes lying in Bohr sets. We extend a theorem of Rogers to the more general real residues setting, and we extend Dirichlet's theorem for prime numbers lying in arithmetic progressions to prime numbers lying in Bohr sets. We also prove that circle rotations equidistribute when sampled along such primes, provided the rotation angle is rationally independent of the Bohr set parameter, generalizing a theorem of Vinogradov. An appendix by Manuel Hauke answers a combinatorial question that is posed in the introduction.

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arXiv2024

The Duffin-Schaeffer conjecture with a moving target

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (\varphi(q)\psi(q)/q)^m = \infty$, we show that for almost every $\mathbf{x} \in\mathbb{R}^m$ there are infinitely many rational vectors $\mathbf{a}/q$ such that $\vert q\mathbf{x} - \mathbf{a} - \mathbf{y}\vert<\psi(q)$ and such that each component of $\mathbf{a}$ is coprime to $q$. This is an inhomogeneous extension of a homogeneous conjecture of Sprind\v{z}uk which was itself proved in 1990 by Pollington and Vaughan. In fact, our main result generalizes Pollington-Vaughan not only to the inhomogeneous case, but also to the setting of moving targets, where the inhomogeneous parameter $\mathbf{y}$ is free to vary with $q$. In contrast, we show by an explicit construction that the (1-dimensional) inhomogeneous Duffin-Schaeffer conjecture fails to hold with a moving target, implying that any successful attack on the one-dimensional problem must use the fact that the inhomogeneous parameter is constant. We also introduce new questions regarding moving targets.

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arXiv2024

Metric bootstraps for limsup sets

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in $m$-dimensional Euclidean space, or a set of $n$-by-$m$ systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in $m$-dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of $n$-by-$m$ systems of linear forms. Consequently, a class of $m$-dimensional results in Diophantine approximation can be bootstrapped to corresponding $n$-by-$m$-dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

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arXiv2023

Inhomogeneous approximation for systems of linear forms with primitivity constraints

We study (inhomogeneous) approximation for systems of linear forms using integer points which satisfy additional primitivity constraints. The first family of primitivity constraints we consider were introduced in 2015 by Dani, Laurent, and Nogueira, and are associated to partitions of the coordinate directions. Our results in this setting strengthen a theorem of Dani, Laurent, and Nogueira, and address problems posed by those same authors. The second primitivity constraints we consider are analogues of the coprimality required in the higher-dimensional Duffin--Schaeffer conjecture, posed by Sprindžuk in the 1970's and proved by Pollington and Vaughan in 1990. Here, with attention restricted to systems of linear forms in at least three variables, we prove a univariate inhomogeneous version of the Duffin--Schaeffer conjecture for systems of linear forms, the multivariate homogeneous version of which was stated by Beresnevich, Bernik, Dodson, and Velani in 2009 and recently proved by the second author.

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arXiv2023

The Duffin--Schaeffer conjecture for systems of linear forms

We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When $m=n=1$, this is the classical 1941 Duffin--Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher-dimensional version, where $m>1$ and $n=1$, in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010 Beresnevich and Velani proved the $m>1$ cases of that. Catlin's classical conjecture, where $m=n=1$, follows from the classical Duffin--Schaeffer conjecture. The remaining cases of the generalized version, where $m=1$ and $n>1$, follow from our main result. Finally, through the Mass Transference Principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich \emph{et al} (2009).

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arXiv2021

Independence inheritance and Diophantine approximation for systems of linear forms

The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb R^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine-Groshev theorem which does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani (2010) that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresnevich, Bernik, Dodson, and Velani (2009), and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin-Schaeffer conjecture. When $nm=1$ it is known by work of Duffin and Schaeffer (1941) that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine-Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine's theorem itself underpins the Khintchine-Groshev theory.

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arXiv2021

Diophantine approximation in metric space

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of `well-spread' points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.

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arXiv2021

A note on sequences not having metric Poissonian pair correlations

The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come arbitrarily close to a threshold below which it is believed that all sequences have MPPC. A similar result appears in work of Lachmann and Technau and is proved using a totally different strategy. The main novelty here is the simplicity of the proof, which we arrive at by modifying a construction of Bourgain.

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arXiv2020

Remarks about inhomogeneous pair correlations

Given an infinite subset $\mathcal A \subseteq\mathbb N$, let $A$ denote its smallest $N$ elements. There is a rich and growing literature on the question of whether for typical $α\in[0,1]$, the pair correlations of the set $αA \pmod 1\subset [0,1]$ are asymptotically Poissonian as $N$ increases. We define an inhomogeneous generalization of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

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arXiv2016

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin--Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szusz's inhomogeneous (1958) version of Khintchine's Theorem (1924). For example, given any real sequence {y_i}, we build a divergent series of non-negative reals psi(n) such that for any y in {y_i}, almost no real number is inhomogeneously psi-approximable with inhomogeneous parameter y. Furthermore, given any second sequence {z_i} not intersecting the rational span of {1,y_i}, and assuming a dynamical version of Erdos' Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously psi-approximable with any inhomogeneous parameter z in {z_i}. Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin--Schaeffer Conjecture and Duffin--Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher's Zero-One Law (1961) for inhomogeneous approximation by reduced fractions.

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arXiv2013

Higher cohomology of parabolic actions on certain homogeneous spaces

We show that for a parabolic R^d-action on a compact quotient of PSL(2,R)^d, the cohomologies in degrees 1 through d-1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers we have established these results for certain Anosov systems. The present work extends the methods of those papers to systems that are not Anosov. The main new idea is in Section 4, where we define special elements of representation spaces that allow us to modify the arguments from the previous papers. In Section 7 we discuss how one may generalize this strategy to R^d-systems coming from a product of Lie groups, like in the systems we have here.

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arXiv2013

Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livshitz Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the "correct" direction in top degree. In Part I we study our "base case": geodesic flows of finite-volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livshitz Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.

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arXiv2012

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2, R) \times \cdot \cdot \cdot \times SL(2, R)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. In the top degree, we show that the obstructions to solving the coboundary equation come from distributions that are invariant under the action. In intermediate degrees, we show that the cohomology trivializes. It has been conjectured by A. and S. Katok that, for a standard partially hyperbolic R^d- or Z^d-action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, in analogy to Livsic's theorem for Anosov flows, and that the intermediate cohomology trivializes, as it is known to do in the first degree, by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. For diagonal subgroup actions on SL(2, R)^d /Γ, our results verify the "intermediate cohomology" part of the conjecture, and are a step in the direction of the "top-degree cohomology" part.

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arXiv2012

Limit theorems for rank-one Lie groups

We investigate asymptotic behaviour of averaging operators for actions of simple rank-one Lie groups. It was previously known that these averaging operators converge almost everywhere, and we establish a more precise asymptotic formula that describes their deviations from the limit.

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arXiv2009

Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows coming from two commuting embedded copies of SL(2,R). In both cases we show that any smooth real-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.

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