arXiv subjects

Felipe A. Ramírez

Publications and source records attributed to Felipe A. Ramírez.

11 recordsLinked to original sources

arXiv2026

The inhomogeneous Khintchine Theorem in dimension two

We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine--Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart.

math.NT↗

arXiv2026

Straight-line trajectories on the Mucube

The dynamics of straight line flows on compact half-translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations possibly composed with rotation by $π$) has been widely studied due to their connections to polygonal billiards and Teichmüller theory. However, much less is known when the underlying surface is non-compact or infinite type. In this paper, we consider the straight line flow of the Mucube -- an infinite $\mathbb{Z}^3$-periodic half-translation square-tiled surface -- first written about by Coxeter and Petrie and more recently studied by Athreya--Lee and Gutiérrez-Romo--Lee--Sánchez. We give a geometric description of the flow's periodic and drift orbits in terms of the Mucube's rigid symmetries, and we give a complete characterization of the set of directions in which the straight line flow is periodic on the Mucube -- first in terms of a genus one quotient and second in terms of an infinitely generated subgroup of $\mathrm{SL}_2(\mathbb{Z})$. We use the latter characterization to obtain the Veech group (i.e. group of derivatives of affine diffeomorphisms) of the Mucube. Finally, we prove density of the sets of periodic and ergodic directions.

math.DS↗

arXiv2025

Twisted approximation with restricted denominators

Given an increasing integer sequence $(a_n)$, a real number $α$, and a sequence $ψ(n)$, we study the set $W$ of real numbers $γ$ for which $a_nα- γ$ is a distance less than $ψ(n)$ away from an integer. This is often referred to as twisted Diophantine approximation, in this case with denominators restricted to the given sequence $(a_n)$. Our main results are about the size of $W$, and they hold for almost every $α$, with respect to a measure of positive Fourier dimension, for example Lebesgue measure. Our results extend recent work of Kristensen and Persson, and answer questions that they posed.

math.NT↗

arXiv2025

Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target

Sz{ü}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$ under which for almost every real number $α$ there exist infinitely many rationals $p/q$ such that \begin{equation*} \lvertα- \frac{p+γ}{q}\rvert < \frac{ψ(q)}{q}, \end{equation*} where $γ\in\mathbb{R}$ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of $qα\,(\bmod 1)$ to a shrinking target centered around $γ\,(\bmod 1)$, viewed in $\mathbb{R}/\mathbb{Z}$. Hauke and the second author have conjectured that Sz{ü}sz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter $γ$ is allowed to depend on the denominator $q$ of the approximating rational. We show that the conjecture holds under an ``extra divergence'' assumption on $ψ$. We also show that it holds when the inhomogeneous parameter's movement is constrained to a finite set. As a byproduct, we obtain a finite-colorings version of the inhomogeneous Khintchine theorem, giving rational approximations with monochromatic denominators.

math.NT↗

arXiv2025

General Duffin--Schaeffer-type counterexamples in diophantine approximation

Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct Duffin--Schaeffer-type counterexamples by restricting the denominator. We also extend these constructions to the inhomogeneous setting. Our results resolve some natural questions arising from the works of Erdős, Vaaler, and Yu.

math.NT↗

arXiv2019

Approximation by random fractions

We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the question of when the monotonicity assumption can be removed. Informally speaking, we show that if the probabilities in our model decay sufficiently fast as the denominator increases, then a Khintchine-like statement holds without a monotonicity assumption. Although our rate of decay of probabilities is unlikely to be optimal, it is known that such a result would not hold if the probabilities did not decay at all.

math.NT↗

arXiv2018

Khintchine's Theorem with random fractions

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schaeffer Conjecture, like the Duffin-Schaeffer Conjecture itself, implies Catlin's Conjecture (1976). It is not obvious whether the Duffin-Schaeffer Conjecture and its random version imply one another, and it is not known whether Catlin's Conjecture implies either of them. The question of whether Catlin implies Duffin-Schaeffer has been unsettled for decades.

math.NT↗

arXiv2018

A characterization of bad approximability

We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate. This implies in particular that Kurzweil's Theorem cannot be restricted to any points in the inhomogeneous part. Our results generalize to weighted approximations, and to higher irrationality exponents.

math.NT↗

arXiv2016

Rational approximation of affine coordinate subspaces of Euclidean space

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the basepoint of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence.

math.NT↗

arXiv2015

Recurrence to shrinking targets on typical self-affine fractals

We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this set is equivalent to the recurring set on the fractal.

math.DS↗

arXiv2015

Khintchine types of translated coordinate hyperplanes

There has been great interest in developing a theory of "Khintchine types" for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes for which there is a dichotomy as in Khintchine's Theorem: the set of rationally approximable points is null or full, according to the convergence or divergence of the series associated to the desired rate of approximation.

math.NT↗