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Let ⟨x⟩ denote the distance from x∈R to the set of integers Z. The Littlewood Conjecture states that for all pairs (α,β)∈R^2 the product q⟨qα⟩⟨qβ⟩ attains values arbitrarily close to 0 as q∈N tends to infinity. Badziahin showed that if a factor logq·loglogq is added to the product, the same statement becomes false. In this paper, we generalise Badziahin’s result to vectors α∈R^d, replacing the function logq·loglogq by (logq)d−1·loglogqfor any d≥2, and thereby obtaining a new proof in the case d=2. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest. 

Let X = G/Γ, where G is a Lie group and Γ is a uniform lattice in G, and let O be an open subset of X. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape O on average with frequency δ, where 0 < δ ≤ 1. 

Recently, the growth of the products of L¨uroth quotients d_i(x) in the L¨uroth expansion of a real number x was studied in connection with improvements to Dirichlet’s theorem. 

Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms. 

This paper is a sequel to [Monatsh. Math. 194 (2021) 523–554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers [Formula: see text] and [Formula: see text], any [Formula: see text], and any homogeneous function [Formula: see text] that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function [Formula: see text] for a generic element [Formula: see text] in the [Formula: see text]-orbit of [Formula: see text] to be (respectively, not to be) [Formula: see text]-approximable at [Formula: see text]: that is, for there to exist infinitely many (respectively, only finitely many) [Formula: see text] such that [Formula: see text] for each [Formula: see text]. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of [Formula: see text] that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace [Formula: see text] above by any closed subgroup of [Formula: see text] that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper. 

The classical Khintchine and Jarnik 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. 

Abstract In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero–one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero–one laws for the two problems. 

In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauß map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co-null set respectively).