arXiv subjects

Ayreena Bakhtawar

Publications and source records attributed to Ayreena Bakhtawar.

13 recordsLinked to original sources

arXiv2026

Sharpening Borel's result in Diophantine approximation

In this paper, we refine Borel's 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients $\Theta_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots]$ as $\Theta_n = q_n^2\left| x-\frac{p_n}{q_n}\right|$. Here $\frac{p_n}{q_n}$ is the $n$th RCF convergent of $x$. Borel's result states that for all (irrational) $x$ and all $n\in\mathbb{N}$, $$ \min \left\{ \Theta_{n-1}(x),\Theta_n(x),\Theta_{n+1}(x)\right\} \leq \frac{1}{\sqrt{5}}. $$ We focus on the situation where $a_{n+1}=1$, since otherwise a result by F.~Bagemihl and J.R.~McLaughlin from 1966 implies that the Borel-bound $1/\sqrt{5}$ can already be improver to $1/\sqrt{8}$.

math.DS↗

arXiv2026

On the minimum of $σ$-Brjuno functions

$σ$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/σ},$ $(σ>0).$ As in the classical case, $B_σ$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_σ$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $σ=n \in \mathbb{N}$, the unique global minimum of $B_n$ is achieved at the fixed point $ [0; \overline{n+1}]$. Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for $σ$ in a neighborhood of $n$. Finally, we discuss the scaling behavior near these minima and we formulate a conjecture about the phase transitions for the location of the minimizer as $σ$ varies.

math.DS↗

arXiv2025

Hausdorff dimension for the weighted products of multiple digits in d-decaying Gauss like systems

We compute the Hausdorff dimension of sets defined by the growth of weighted products of multiple digits at arbitrary positions in $d$-decaying Gauss-like iterated function systems. We provide the complete Hausdorff dimensional result for product of more than two digits, which was an open problem even for consecutive digits in the classical Gauss map and Lüroth map. In our approach we do not need to assume the Bounded Distortion Property (BDP).

math.DS↗

arXiv2025

Sharpening Vahlen's result in Diophantine approximation

n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients $a_n$ or $a_{n+1}$ of the regular continued fraction expansion $[a_0;a_1,a_2,\dots]$ of $x$ is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Hanucl ([9]), Hanucl and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Hanucl c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant.

math.DS↗

arXiv2025

Uniform Diophantine approximation on the Hecke group $\mathbf H_4$

Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_4$. For a given real number $α$, we characterize the sequence of $\mathbf H_4$-best approximations of $α$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $α$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.

math.NT↗

arXiv2025

Global and local minima of $α$-Brjuno functions

The main goal of this article is to analyze some peculiar features of the global (and local) minima of $α$-Brjuno functions $B_α$ where $α\in(0,1].$ Our starting point is the result by Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at $g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$ we shall deduce that all preimages of $g$ under the Gauss map are also local minima for $B_1$. Next we consider the problem of characterizing global and local minima of $B_α$ for other values of $α$: we show that for $α\in (g,1)$ the global minimum is again attained at $g$, while for $α$ in a neighbourhood of $1/2$ the function $B_α$ attains its minimum at $γ:=\sqrt{2}-1$. The fact that the minimum of $B_α$ is attained when $α$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_α$ is lower semicontinuous for all rational $α,$ but we also exhibit an irrational $α$ for which $B_α$ is not lower semicontinuous. %We also prove that if $α$ is rational then $B_α$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_α$ is not lower semicontinuous for a suitable irrational $α.$

math.DS↗

arXiv2024

Regularity properties of the $α$-Wilton functions

The aim of this article is to study the regularity properties of the Wilton functions $W_α$ associated with $α$-continued fractions. We prove that the Wilton function is BMO for $α\in[1-g,g]$ (where $g:=\frac{\sqrt{5}-1}{2}$ denotes the golden number), and we show that this result is optimal, since we find that on any left neighbourhood of $1-g$ and on any right neighbourhood of $g$ there are values $α$ for which $W_α$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $α$-continued fractions called ``matching''. Our results complete those of Marmi--Moussa--Yoccoz (1997) and of Lee--Marmi--Petrykiewicz--Schindler (2024), where it is proven that Wilton function is BMO for, respectively, $α=1/2$ (\cite{MaMoYo_97}) and $α\in[\frac{1}{2},g]$ (\cite{LeMar_24}).

math.DS↗

arXiv2023

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

Let $ψ:\mathbb R_{+}\to \mathbb R_{+}$ 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 $ψ$-Dirichlet improvable, if the system $$\|A\mathbf q +\mathbf b-\mathbf p\|^m<ψ(T), \quad \|\mathbf q\|^n<T$$ 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 $ψ$-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 $ψ$-Dirichlet non-improvable sets.

math.NT↗

arXiv2022

Increasing rate of weighted product of partial quotients in continued fractions

Let $[a_1(x),a_2(x),\cdots,a_n(x),\cdots]$ be the continued fraction expansion of $x\in[0,1)$. In this paper, we study the increasing rate of the weighted product $a^{t_0}_n(x)a^{t_1}_{n+1}(x)\cdots a^{t_m}_{n+m}(x)$ ,where $t_i\in \mathbb{R}_+\ (0\leq i \leq m)$ are weights. More precisely, let $φ:\mathbb{N}\to\mathbb{R}_+$ be a function with $φ(n)/n\to \infty$ as $n\to \infty$. For any $(t_0,\cdots,t_m)\in \mathbb{R}^{m+1}_+$ with $t_i\geq 0$ and at least one $t_i\neq0 \ (0\leq i\leq m)$, the Hausdorff dimension of the set $$\underline{E}(\{t_i\}_{i=0}^m,φ)=\left\{x\in[0,1):\liminf\limits_{n\to \infty}\dfrac{\log \left(a^{t_0}_n(x)a^{t_1}_{n+1}(x)\cdots a^{t_m}_{n+m}(x)\right)}{φ(n)}=1\right\}$$ is obtained. Under the condition that $(t_0,\cdots,t_m)\in \mathbb{R}^{m+1}_+$ with $0<t_0\leq t_1\leq \cdots \leq t_m$, we also obtain the Hausdorff dimension of the set \begin{equation*} \overline{E}(\{t_i\}_{i=0}^m,φ)=\left\{x\in[0,1):\limsup\limits_{n\to \infty}\dfrac{\log \left(a^{t_0}_n(x)a^{t_1}_{n+1}(x)\cdots a^{t_m}_{n+m}(x)\right)}{φ(n)}=1\right\}.\end{equation*}

math.NT↗

arXiv2022

Metrical properties for the weighted products of multiple partial quotients in continued fractions

The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set: $$ {\D_{\mathbf t}}(ψ):=\left\{x\in[0, 1): \prod\limits_{i=0}^{m-1}{a^{t_i}_{n+i}(x)} \ge Ψ(n)\ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $t_i\in\mathbb R_+$ for all ${0\leq i\leq m-1}$, and $Ψ:\N\to\R_{\ge 1}$ is a positive function.

math.NT↗

arXiv2020

Hausdorff dimension for the set of points connected with the generalized Jarník-Besicovitch set

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in\mathbb{N},$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{τ(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*} holds for infinitely many $n\in\mathbb{N},$ where $h$ and $τ$ are positive continuous functions, $T$ is the Gauss map and $a_n(x)$ denote the $n$th partial quotient of $x$ in its continued fraction expansion. By appropriate choices of $r$, $τ(x)$ snd $h(x)$ we obtain the classical Jarník-Besicovitch Theorem as well as more recent results by Wang-Wu-Xu, Wang-Wu, Huang-Wu-Xu and Hussain-Kleinbock-Wadleigh-Wang.

math.NT↗

arXiv2019

Hausdorff dimension of a set in the theory of continued fractions

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.

math.DS↗

arXiv2019

The sets of Dirichlet non-improvable numbers vs well-approximable numbers

Let $Ψ:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$'{th} partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$'{th} convergent. The set of $Ψ$-Dirichlet non-improvable numbers \begin{equation*} G(Ψ):=\Big\{x\in \lbrack 0,1):a_{n}(x)a_{n+1}(x)\,>\,Ψ\big(q_{n}(x) \big)\ \mathrm{for\ infinitely\ many}\ n\in \mathbb{N}\Big\}, \end{equation*} is related with the classical set of $1/q^{2}Ψ(q)$-approximable numbers $ \mathcal{K}(Ψ)$ in the sense that $\mathcal{K}(3Ψ)\subset G(Ψ)$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(Ψ)\setminus \mathcal{K}(3Ψ)$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets $\mathcal{K}(Ψ)$ and $G(Ψ)$. This gives an affirmative answer to a question raised by Hussain-Kleinbock-Wadleigh-Wang (2018).

math.NT↗