arXiv subjects

David Simmons

Publications and source records attributed to David Simmons.

At least 19 recordsLinked to original sources

arXiv2026

Well and badly approximable sets, and rapid winning

The set of $\tau$-approximable numbers, $\mathcal W(\tau)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^\gamma$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $\Psi$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $\psi$, we prove that $\mathcal W(\psi)\cap\Bad^\gamma$ is strong $\Psi$-rapid winning for a natural gauge $\Psi$ determined by $\psi$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $\Psi$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $\Psi$. In particular, for $\psi(q)=q^{-\tau}, \tau\ge1,$ we recover the exact Jarn\'ik--Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(\tau)\cap\Bad^\gamma\bigr)=\frac{2}{\tau+1}.$$

math.NT↗

arXiv2026

Gradualist descriptionalist set theory

We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each $k \in \mathbb N$ of a sequence of ordinals $\eta_0 < . . . < \eta_k$ such that for each $i < k$, $\eta_i$ is $\eta_{i+1}$-reflecting, a notion we introduce which implies being $\Pi_n$-reflecting for all $n \in \mathbb N$ (and in particular being admissible and recursively Mahlo).

math.LO↗

arXiv2025

McMullen's game for equicontinuously-twisted badly approximable points in continued fractions and beta expansions

In a beta-transformation (for integer beta) or a Gauss map system, given a sequence of functions fn from [0,1] to itself, consider the collection of points in [0,1] whose nth iteration under the map is distanced away from its value under fn. It is well known that for constant sequences fn, such collections are always winning in McMullen's game and in particular they have Hausdorff dimension 1. We extend the results to all equicontinuous sequences of functions fn.

math.DS↗

arXiv2025

Twisted Diophantine approximation on manifolds

In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\boldsymbol\alpha$ one is interested in sets of vectors $\boldsymbol\beta\in\mathbb R^m$ such that the system of affine forms $\mathbb R^n \ni \mathbf q \mapsto \boldsymbol\alpha\mathbf q + \boldsymbol\beta \in \mathbb R^m$ satisfies some given Diophantine condition. In this paper we introduce the notion of manifolds which are of $\boldsymbol\alpha$-twisted Khintchine type for convergence or divergence. We provide sufficient conditions under which nondegenerate analytic manifolds exhibit this twisted Khintchine-type behaviour. Furthermore, we investigate the intersection properties of the sets of $\boldsymbol\alpha$-twisted badly approximable and well approximable vectors with nondegenerate manifolds.

math.NT↗

arXiv2025

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abelian variety is equivalent to a well-known Diophantine condition holding for a certain matrix related to that variety. We then posit a related but weaker conjecture, and establish the upper bound direction of that conjecture in full generality. For rank 1 elliptic curves defined over a number field $K \subset \mathbb{R}$, we then obtain a weak-type Dirichlet theorem in this setting, establish the optimality of this statement, and prove our conjecture in this case.

math.NT↗

arXiv2024

Hausdorff dimension of differences of badly approximable sets

The set of badly approximable numbers, Bad, is known to be winning for Schmidt's game and hence has full Hausdorff dimension. It is also known that the set of inhomogeneously badly approximable numbers has full dimension. We prove that the set difference also has full dimension using a variant of the Schmidt game, which we call the rapid game, played on the space of unimodular grids.

math.NT↗

arXiv2024

Shrinking targets versus recurrence: the quantitative theory

Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[ R(x,N;T,\psi) := \#\{1\leq n\leq N: d(T^n x, x) < \psi(n)\}. \] We show that for any $\varepsilon > 0$ we have \[ R(x,N;T,\psi) = \Psi(N)+O\left(\Psi^{1/2}(N) \ (\log\Psi(N))^{3/2+\varepsilon}\right) \] for $\mu$-almost all $x\in X$ and for all $N\in\mathbb N$, where $\Psi(N):= 2 \sum_{n=1}^N \psi(n)$. We also prove a generalization of this result to higher dimensions.

math.DS↗

arXiv2023

Exact dimensions of the prime continued fraction Cantor set

We study the exact Hausdorff and packing dimensions of the $prime$ $Cantor$ $set$, $\Lambda_P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi(r) = r^\delta \log^{-2\delta}\log(1/r)$, where $\delta$ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

math.NT↗

arXiv2022

Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions

Let $\psi:\mathbb R_{+}\to \mathbb R_{+}$ be a nonincreasing function. A pair $(A,\mathbf b),$ where $A$ is a real $m\times n$ matrix and $\mathbf b\in\mathbb R^{m},$ is said to be $\psi$-Dirichlet improvable, if the system $$\|A\mathbf q +\mathbf b-\mathbf p\|^m<\psi(T), \quad \|\mathbf q\|^n<T$$ is solvable in $\mathbf p\in\mathbb Z^{m},$ $\mathbf q\in\mathbb Z^{n}$ for all sufficiently large $T$ where $\|\cdot\|$ denotes the supremum norm. For $\psi$-Dirichlet non-improvable sets, Kleinbock--Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim--Kim (2021) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff $f$-measure version of Kim--Kim (2021) results for $\psi$-Dirichlet non-improvable sets.

math.NT↗

arXiv2020

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large enough $t$. Denote the collection of such points by $D(\psi)$. In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sub-linear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang (2018), for some non-essentially sub-linear dimension functions, and for all dimension functions but with a growth condition on the approximating function.

math.NT↗

arXiv2020

Dynamical Borel-Cantelli lemma for recurrence theory

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $\mu$-measure of the following set \[ R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $\psi:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.

math.DS↗

arXiv2020

Regularity of Almost-Minimizers of H\"older-Coefficient Surface Energies

We study almost-minimizers of anisotropic surface energies defined by a H\"older continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally H\"older continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.

math.AP↗

arXiv2020

Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism \`a la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarn\'ik and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\leq N$; as well as its counterpart for reals whose partial quotients are all $\geq N$ due to Good from 1941.

math.DS↗

arXiv2020

Metrical theorems on systems of affine forms

In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by $\qq\mapsto \qq X+\bfalpha$, where $\qq\in\Z^m$ (viewed as a row vector), $X$ is an $m\times n$ real matrix and $\bfalpha\in \R^n$. The classical setting refers to the ${\rm dist}(\qq X+\bfalpha, \Z^m)$ to measure the closeness of the integer values of the system $(X, \bfalpha)$ to integers. The absolute value setting is obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, \0)$; and the more general mixed settings are obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, Λ)$, where $Λ$ is a subgroup of $\Z^m$. We prove the Khintchine--Groshev and Jarník type theorems for the mixed affine forms and Jarník type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

math.NT↗

arXiv2019

The Measure Game

We study a game first introduced by Martin (actually we use a slight variation of this game) which plays a role for measure analogous to the Banach-Mazur game for category. We first present proofs for the basic connections between this game and measure, and then use the game to prove fundamental measure theoretic results such as Fubini's theorem, the Borel-Cantelli lemma, and a general unfolding result for the game which gives, for example, the measurability of $\boldsymbolΣ^1_1$ sets. We also use the game to give a new, more constructive, proof of a strong form of the Rényi-Lamperti lemma, an important result in probability theory with many applications to number theory. The proofs we give are all direct combinatorial arguments using the game, and do not depend on known measure theoretic arguments.

math.LO↗

arXiv2019

On the dimension spectra of infinite iterated function systems

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions -- (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect -- which resolve questions posed by Chousionis, Leykekhman, and Urba\'nski, and go on provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.

math.DS↗

arXiv2019

Schmidt's game on Hausdorff metric and function spaces: generic dimension of sets and images

We consider Schmidt's game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behaviour of objects in a metric space, mostly in the context of fractal dimensions, and the notion of `generic' we adopt is that of being winning for Schmidt's game. We find properties whose corresponding sets are winning for Schmidt's game that are starkly different from previously established, and well-known, properties which are generic in other contexts, such as being residual or of full measure.

math.MG↗

arXiv2019

Geometric Limits of Julia Sets with Parameters on the Circle

We show that the geometric limit as $n \rightarrow \infty$ of the filled Julia sets $K(P_{n,c})$ for the maps $P_{n,c}(z) = z^n + c$ does not exist for almost every $c$ on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle, and this is used to show that for certain parameters, the geometric limit of the Julia sets $J(P_{n,c})$ is the unit circle.

math.DS↗