Let C be two times continuously differentiable curve in R^2 with at least one point at which the curvature is non-zero. For any i,j > 0 with i+j =1, let Bad(i,j) denote the set of points (x,y) in R^2 for which max {||qx ||^{1/i}, ||qy||^{1/j}} > c/q for all integers q >0. Here c = c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.
Given an $a$-dimensional linear subspace $\mathfrak{A}$ in $\mathbb{R}^d$ which contains a badly approximable $b$-dimensional subspace $\mathfrak{B} \subset \mathfrak{A}$. We study the badly approximability almost all $c$-dimensional linear subspaces $\mathfrak{C} \subset \mathfrak{A}$.
We provide a number of conditions on the rational numbers $u$ and $v$ which ensure that the Laurent series $g_{u,v}(x):=\prod_{t=0}^\infty (1+ux^{-3^t} + vx^{-2\cdot 3^t})$ is badly approximable.
We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong form of a result by Haynes, Jensen and Kristensen. Finally, we establish a similar result involving inhomogeneously badly approximable numbers, making progress towards a problem posed by Pollington, Velani, Zafeiropoulos and Zorin.
We consider approximation of vectors $\mathbf{z}\in F\otimes\mathbb{R}\cong\mathbb{R}^r\times\mathbb{C}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of $SL_2(\mathcal{O}_F)\backslash SL_2(F\otimes\mathbb{R})$ naturally associated to (totally indefinite, anisotropic) $F$-rational binary quadratic and Hermitian forms, a generalization of the well-known fact that quadratic irrationals are badly approximable over $\mathbb{Q}$.
Motivated by a wonderful paper by Ngoc Ai Van Nguyen, Anthony Poëls and Damien Roy, where a powerful method was introduced, we prove a criterion for a vector $\pmbα\in \mathbb{R}^d$ to be a badly approximable vector. Moreover we construct certain examples which show that a more general version of our criterion is not valid.
我们证明了Thue-Morse常数$τ_{TM}=0.01101001..._2$不是一个差逼近数。此外，我们还证明了$τ_{TM}(a)=0.01101001..._a$对每个整数基$a\geq 2$都不是差逼近的，使得$a$不能被15整除.同时给出了洛朗级数$\tilde{f}_{TM}(z) = z^{-1}\prod_{n=1}^\infty (1-z^{-2^n})$的一个精确的换算公式，进一步发展了Alf van der Poorten等人的研究.
In this paper, we study the weighted $n$-dimensional badly approximable points on manifolds. Given a $C^n$ differentiable non-degenerate submanifold $\mathcal{U} \subset \mathbb{R}^n$, we will show that any countable intersection of the sets of the weighted badly approximable points on $\mathcal{U}$ has full Hausdorff dimension. This strengthens a result of Beresnevich by removing the condition on weights and weakening the smoothness condition on manifolds. Compared to the work of Beresnevich, our approach relies on homogeneous dynamics. It turns out that in order to solve this problem, it is …
In this paper we investigate the problem of how well points in finite dimensional p-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet's theorem and the Duffin-Schaeffer theorem. We prove a complementary result, showing that the set of badly approximable points has maximum Hausdorff dimension. Our proof is a simple application of the elegant machinery of Schmidt's game.
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common approach to studying the continued fraction expansions of zeroes of binary forms, via consideration of the action of the general linear group, and apply it to discuss expansions of quadratic surds on the one hand and of badly approximable numbers on the other hand. Also, we generalize the property of the simple continued fraction expansions that the convergents a…
We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n θ$, in the case where $θ$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.
Given a badly approximable number $α$, we study the asymptotic behaviour of the Sudler product defined by $P_N(α) = \prod_{r=1}^N 2 | \sin πr α|$. We show that $\liminf_{N \to \infty} P_N(α) = 0$ and $\limsup_{N \to \infty} P_N(α)/N = \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds $7$ infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value $7$ is optimal, even when restrictin…
Cycle integrals of modular functions are expected to play a role in real quadratic analogue of singular moduli. In this paper, we extend the definition of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers. We also give explicit representations of values of extended-cycle integrals for some cases.
In this article we introduce the notion of badly approximable matrices of higher order using higher sucessive minima in $\mathbb R^d$. We prove that for order less than $d$, they have Lebesgue measure zero and the gaps between them still have full Hausdorff dimension.
In this paper we prove that badly approximable points on any analytic non-degenerate curve in $\mathbb{R}^n$ is an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani (2014) which represents a far-reaching generalisation of Davenport's problem from the 1960s. Amongst various consequences of our main result is a solution to Bugeaud's problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.
We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields $K$ and characterize the "badly approximable" numbers, ($z$ such that there is a $C(z)>0$ with $|z-p/q|\geq C/|q|^2$ for all $p/q\in K$), by boundedness of the partial quotients in the continued fraction expansion of $z$. Applying this algorithm to "tagged" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in $\mathbb{C}$ whose points are badly approximable over $K$, with effective constants. By other methods (the Dani correspondence), we prove the exi…
We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular on average. We show that if $\dim_H Bad_A(ε)=m$, then $A$ is singular on average.
For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approximable matrices for fixed target $b$ and the set of $ε$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $ε$-badly appr…
For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) \in R^2 such that max \{ ||qx||^{1/i}, \, ||qy||^{1/j} \} > c/q for some positive constant c = c(x,y) and all q in N. We show that \Bad(i,j) \cap C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we o…
