Khintchine Type Research Articles

A Cramér–Wold device for infinite divisibility of Zd-valued distributions

We show that a Cramér–Wold device holds for infinite divisibility of Zd-valued distributions, i.e. that the distribution of a Zd-valued random vector X is infinitely divisible if and only if the distribution of aTX is infinitely divisible for all a∈Rd, and that this in turn is equivalent to infinite divisibility of the distribution of aTX for all a∈N0d. A key tool for proving this is a Lévy–Khintchine type representation with a signed Lévy measure for the characteristic function of a Zd-valued distribution, provided the characteristic function is zero-free.

S-arithmetic inhomogeneous Diophantine approximation on manifolds

We prove S-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For S consisting of more than one valuation, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the S-arithmetic inhomogeneous counterpart of the Sprindžuk conjectures using nondivergence estimates for flows on homogeneous spaces in conjunction with the transference principle of Beresnevich and Velani.

On the multiple recurrence properties for disjoint systems

We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space \((X,{\cal B},\mu )\). We obtain the multiple recurrence property of T1, …, Tk and this result is utilized to derive multiple recurrence of Poincaré type in metric spaces. We also present the multiple recurrence property of Khintchine type. Further, we study multiple ergodic averages of disjoint systems and we show that T1, …, Tk are uniformly jointly ergodic if each Ti is ergodic.

A Khintchine-type theorem for affine subspaces

We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of [Formula: see text] intersected with an appropriate subspace is also obtained.

Graph product Khintchine inequalities and Hecke C⁎-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness

Graph products of groups were introduced by Green in her thesis [32]. They have an operator algebraic counterpart introduced and explored in [14]. In this paper we prove Khintchine type inequalities for general C⁎-algebraic graph products which generalize results by Ricard and Xu [50] on free products of C⁎-algebras. We apply these inequalities in the context of (right-angled) Hecke C⁎-algebras, which are deformations of the group algebra of Coxeter groups (see [22]). For these we deduce a Haagerup inequality which generalizes results from [33]. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C⁎-algebras. Lastly we characterize exactness and nuclearity of general Hecke C⁎-algebras.

On a Counting Theorem for Weakly Admissible Lattices

Abstract We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve “hyperbolic spikes” and occur naturally in multiplicative Diophantine approximation. We use Wilkie’s o-minimal structure $\mathbb{R}_{\exp }$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The 1st one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The 2nd application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result, we develop a sophisticated partition method that is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation

There are two main interrelated goals of this paper. Firstly we investigate the sums $S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}$ and $R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|},$ where $\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W.M.Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of $\{1\le n\le N:\|n\alpha-\gamma\|<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.

Diophantine approximation on the parabola with non-monotonic approximation functions

AbstractWe show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.

An analogue of Khintchine's theorem for self-conformal sets

AbstractKhintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

On Khintchine type inequalities for [formula omitted]-wise independent Rademacher random variables

We consider Khintchine type inequalities on the pth moments of vectors of Nk-wise independent Rademacher random variables. We show that an analogue of Khintchine’s inequality holds, with a constant N1∕2−k∕2p, when k is even. We then show that this result is sharp for k=2; in particular, a version of Khintchine’s inequality for sequences of pairwise Rademacher variables cannot hold with a constant independent of N. We also characterize the cases of equality and show that, although the vector achieving equality is not unique, it is unique (up to law) among the smaller class of exchangeable vectors of pairwise independent Rademacher random variables. As a fortunate consequence of our work, we obtain similar results for 3-wise independent vectors.

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.

Restricted Khinchine Inequality

Abstract We prove a Khintchine type inequality under the assumption that the sumof Rademacher randomvariables equals zero. We also showa newtail-bound for a hypergeometric random variable.

A Contribution to Metric Diophantine Approximation: the Lebesgue and Hausdorff Theories

This thesis is concerned with the theory of Diophantine approximation from the point of view of measure theory. After the prolegomena which conclude with a number of conjectures set to understand better the distribution of rational points on algebraic planar curves, Chapter 1 provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is to develop a theory of approximation in the so far unstudied case when such curves are not defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities. In short, in this type of problem, one prescribes the set of integers which the denominators of all the possible rational approximants of a given vector have to belong to. Finally, a reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This provides the first example of a Khintchine type result in the context of so–called uniform approximation.

A Cantor set type result in the field of formal Laurent series

We prove a Khintchine type theorem for approximation of elements in the Cantor set, as a subset of the formal Laurent series over $\mathbb{F}_3$, by rational functions of a specific type. Furthermore we construct elements in the Cantor set with any prescribed irrationality exponent $\geq 2$.

Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.

Lévy–Khintchine type representation of Dirichlet generators and semi-Dirichlet forms

Abstract Let U be an open set of ℝ n , m be a positive Radon measure on U such that supp [ m ] = U ${{\rm supp}[m]=U}$ , and ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ be a strongly continuous contraction sub-Markovian semigroup on L 2 ( U ; m ) ${L^2(U;m)}$ . We investigate the structure of ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ . (i) Denote respectively by ( A , D ( A ) ) ${(A,D(A))}$ and ( A ^ , D ( A ^ ) ) ${(\hat{A},D(\hat{A}))}$ the generator and the co-generator of ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ . Under the assumption that C 0 ∞ ( U ) ⊂ D ( A ) ∩ D ( A ^ ) ${C^{\infty }_0(U)\subset D(A)\cap D(\hat{A})}$ , we give an explicit Lévy–Khintchine type representation of A on C 0 ∞ ( U ) ${C^{\infty }_0(U)}$ . (ii) If ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form ( ℰ , D ( ℰ ) ) ${({\cal E}, D({\cal E}))}$ , we give an explicit characterization of ℰ on C 0 ∞ ( U ) ${C^{\infty }_0(U)}$ under the assumption that C 0 ∞ ( U ) ⊂ D ( ℰ ) ${C^{\infty }_0(U)\subset D({\cal E})}$ . We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.

The arithmetic of distributions in free probability theory

Abstract We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I 0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I 0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.

Operator spaces and Araki-Woods factors: A quantum probabilistic approach

We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasi-free state. Our approach yields a Khintchine type inequality for the q-gaussian variables for all values q between -1 and 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case.

The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Ito’s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy–Khintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein–Kantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E ( M ) be the associated noncommutative function space. Let ( ε k ) k ⩾ 1 be a Rademacher sequence, on some probability space Ω. For finite sequences ( x k ) k ⩾ 1 of E ( M ) , we consider the Rademacher averages ∑ k ε k ⊗ x k as elements of the noncommutative function space E ( L ∞ ( Ω ) ⊗ ¯ M ) and study estimates for their norms ‖ ∑ k ε k ⊗ x k ‖ E calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖ ∑ k ε k ⊗ x k ‖ E is equivalent to the infimum of ‖ ( ∑ y k ∗ y k ) 1 2 ‖ + ‖ ( ∑ z k z k ∗ ) 1 2 ‖ over all y k , z k in E ( M ) such that x k = y k + z k for any k ⩾ 1 . Dual estimates are given when E is 2-convex and has a nontrivial upper Boyd index. In this case, ‖ ∑ k ε k ⊗ x k ‖ E is equivalent to ‖ ( ∑ x k ∗ x k ) 1 2 ‖ + ‖ ( ∑ x k x k ∗ ) 1 2 ‖ . We also study Rademacher averages ∑ i , j ε i ⊗ ε j ⊗ x i j for doubly indexed families ( x i j ) i , j of E ( M ) .