Let $E\subset [0,1]$ be a set that supports a probability measure $μ$ with the property that $|\widehatμ(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension.
It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coi…
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.
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 measur…
In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.
In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'ık, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.
We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.
For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted' inhomogeneous analogue of Bad(j_1,...,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j_i=1/n, and in the weighted setting when x is chosen from Bad(j_1,...,j_n). We generalise these results proving the full Hausdorff dimension in the weighted se…
For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x:=(x_1, x_2) in R^2 for which there exists a positive constant c(x) such that max {||qx_1||^(-i), ||qx_2||^(-j)} > c(x)/q for all q in N. A new characterization of the set Bad(i, j) in terms of `well-approximable' vectors in the area of 'twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad^x(i, j), the `twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen…
The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider three inhomogeneous problems that further develop these classical results. Firstly, we obtain a Jarnik type theorem for the set of multiplicatively approximable points in the plane. This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting. Next, we establish a Ja…
Let (X,d) be a metric space and (Ω, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ωwhich `stay clear' of some given set of points in X. The classical set \Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets \Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ωhave full Hausdorff dimension. Applications of o…
We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalizing to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorff $f$-measure of the set of matrices which are not $ψ$-approximable, given a dimension function $f$ and a function $ψ:(0,\infty)\to (0,\infty)$. This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff $f$-…
I consider the Diophantine approximation problem of sup-norm simultaneous rational approximation with common denominator of a pair of irrational numbers, and compute explicitly some pairs with large approximation constant. One of these pairs is the most badly approximable pair yet computed.
This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the notion of badly approximability.
The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well approximable points lying on a curve which is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the d…
A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of $ψ$-approximable points in the plane. The construction and use of Haudsorff measure and dimension are explained and the notion of ubiquity, which is effective in estimating the lower bound of the Hausdorff dimension for quite general lim sup sets, is described. An application is made to obt…
The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs …
We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine (1926) and to an original method invented by Y. Peres and W. Schlag (2001).
A classical result of Kaufman states that, for each $τ>1,$ the set of well approximable numbers \[ E(τ)=\{x\in\mathbb{R}: \|qx\| < |q|^{-τ} \text{ for infinitely many integers q}\} \] is a Salem set with Hausdorff dimension $2/(1+τ)$. A natural question to ask is whether the same phenomena holds for well approximable vectors in $\mathbb{R}^n.$ We prove that this is in general not the case. In addition, we also show that in $\mathbb{R}^n, n\geq 2,$ the set of badly approximable vectors is not Salem.
In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on $\mathbb{R}^2$ and prove that, for each such norm, the set of Dirichlet improvable pairs contains the set of badly approximable pairs, hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk--Rogers and Mahler to establish a Hajós--Minkowski type result for the critical locus of a cylinder. As a corollary, using a recent res…
We initiate the study of an intrinsic notion of Diophantine approximation on a rational Carnot group $G$. If $G$ has Hausdorff dimension $Q$, we show that its Diophantine exponent is equal to $(Q+1)/Q$, generalizing the case $G=\mathbb R^n$. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group $\mathbb H^n$, distinguishing between two notions of Diophantine approximation by rational points in $\mathbb H^n$: Carnot Diophantine approximation and Siegel Diophantine approximation. After computing the Siegel Diophantin…
In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.
Let $Γ= Z A +Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let $ω(A)$ denote the exponent of uniform simultaneous rational approximation to the point $A$. We show that for any real number $v\ge ω(A)$, the Hausdorff dimension of the set $B_v$ of points in $R^n$ which are $v$-approximable with respect to $Γ$, is equal to $1/v$.
Let Q be an infinite set of positive integers. Denote by W(Q) the set of n-tuples of real numbers simultaneously tau-well approximable by infinitely many rationals with denominators in Q but only by finitely many rationals with denominators in the complement of Q. The Hausdorff dimension of the liminf set W(Q) is computed when tau > 2 + 1/n. A p-adic analogue of the problem is also studied.
We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $ψ$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for algebraically "nice" manifolds. This generalizes earlier work done in the case of curves.
This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of the sets of weighted badly approximable points. The problems have been recently settled in dimension two but remain open in higher dimensions. In this paper we develop new techniques that allow us to tackle them in full generality. The techniques rest on lattice points counting and a powerful quantitative result of Bernik, Kleinbock and Margulis. The main theo…
This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdor? dimension results for the sets of simultaneously well-approximable points on planar curves, established in [2, 6, 10, 20].
Let $Y_0$ be a not very well approximable $m\times n$ matrix, and let $M$ be a connected analytic submanifold in the space of $m\times n$ matrices containing $Y_0$. Then almost all $Y\in M$ are not very well approximable. This and other similar statements are cast in terms of properties of certain orbits on homogeneous spaces and deduced from quantitative nondivergence estimates for `quasi-polynomial' flows on on the space of lattices.
We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent" for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature pr…
The inhomogeneous metric theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is developed. Our results naturally incorporate the homogeneous Khintchine-Jarnik type theorems recently established in [Ann. of Math. (2), 166 (2007), pp. 367-426] and [Invent. Math., 166 (2006), pp. 103-124]. The key lies in obtaining essentially the best possible results regarding the distribution of `shifted' rational points near planar curves.
The convergence theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the distribution of rational points' (by V. Beresnevich, H. Dickinson, and S. Velani, with an Appendix: `Sums of two squares near perfect squares' by R.C. Vaughan, To appear: Annals of Math., Pre-print: arkiv:math.NT/0401148, (2004), 1-52.) and thereby completes the general metric theory for planar curves.
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $ζ_{1},ζ_{2},...,ζ_{k}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\geq 2$) of $ζ_{1},ζ_{2},...,ζ_{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $ζ_{1},ζ_{2},...,ζ_{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.
The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law f…
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no …
The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the Jarník-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. Th…
Given $n\in\mathbb{N}$ and $τ>\frac1n$, let $\mathcal{S}_n(τ)$ denote the classical set of $τ$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-τ-1}$ from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $τ>\frac1n$ almost all points on $\mathcal{M}$ are not $τ$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the H…
Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $Ψ$-approximable points on nondegenerate manifolds. The problem relates the `size' of the set of $Ψ$-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any nondegenerate manifold. The convergence case, however, represents a major challenging open problem and progress thus far has been effectuated i…
For a given rotation number we compute the Hausdorff dimension of the set of well approximable numbers. We use this result and an inhomogeneous version of Jarnik's theorem to show strong recurrence properties of the billiard flow in certain polygons
Let Q be an infinite set of positive integers. Denote by W_{τ, n}(Q) (resp. W_{τ, n}) the set of points in dimension n simultaneously τ--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial lower bound for the Hausdorff dimension of the liminf set W_{τ, n}\W_{τ, n}(Q) is established when n>1 and τ>1+1/(n-1) in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension as well as the one--dimensional analogue of the problem are also discussed.
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 base point of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence. We also prove a partial analogue regarding the Hausdorff measure theory. Furthermore, we obtain various results relating weighted Diophantine approximation and Dirichlet improvability. In particu…
Let $\cS_n(ψ_1,...,ψ_n)$ denote the set of simultaneously $(ψ_1,...,ψ_n)$--approximable points in $\R^n$ and $\cSM_n(ψ)$ denote the set of multiplicatively $ψ$--approximable points in $\R^n$. Let $\cM$ be a manifold in $\R^n$. The aim is to develop a metric theory for the sets $ \cM \cap \cS_n(ψ_1,...,ψ_n) $ and $ \cM \cap \cSM_n(ψ) $ analogous to the classical theory in which $\cM$ is simply $\R^n$. In this note, we mainly restrict our attention to the case that $\cM$ is a planar curve $\cC$. A complete Hausdorff dimension theory is established for the sets $ \cC \cap \cS_2(ψ_1,ψ_2) $ and $ \…
We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(α,β)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.
Under the assumption that the approximating function $ψ$ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $ψ$-approximable matrices in $\R^{mn}$. The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on $ψ$ is absolutely necessary when $m=n=1$. On the other hand, it is known that monotonicity is not necessary when $n > 2$ (Sprindzuk) or when $n=1$ and $m>1$ (Gallagher). Surprisingly, when $n=2$ the situation is unresolved. We deal with this remaining case and thereby remove all …
We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …
We show that Mahler's classification of real numbers $ζ$ with respect to the growth of the sequence $(w_{n}(ζ))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(λ_{n}(ζ))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximatio…
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also establish the convergence theory for the set of simultaneously $ψ$-approximable points lying on a generic hypersurface, thereby settling the generalized Baker-Schmidt problem in the simultaneous setting for generic hypersurfaces. These results are obtained as consequences of an optimal upper bound for the density of rational points near manifolds of the form $\{ (…
Let $W(\p)$ denote the set of $\p$-well approximable points in $\R^d$ and let $K$ be a compact subset of $\R^d$ which supports a measure $μ$. In this short note, we show that if $μ$ is an `absolutely friendly' measure and a certain $μ$--volume sum converges then $μ(W(\p) \cap K) = 0$. The result obtained is in some sense analogous to the convergence part of Khintchines classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced by D. Kleinbock, E. Lindenstrauss and B. Weiss (On fractal measures…
We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.
For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criter…
The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square $[0,1]^2$ exhibits superdensity, which is a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. In this paper, we give a proof of the extension of this result of Khinchin to the case when the unit torus $[0,1]^2$ is replaced by a finite polysquare surface, or square tiled surface. The argument is based on diophantine approximation and continued fractions, traditional tools in number theory. In particular, we use the famo…
