Related papers: Weighted Diophantine approximation on manifolds
In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.
The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in $R^n$ akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the…
Almost uniform version of noncommutative Wiener-Wintner ergodic theorem and its extension to Besicovitch weights are proved.
We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called "non-cancellation property" which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their…
The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main…
We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly…
It is known that the properties of almost all points of R^n being not very well (multiplicatively) approximable are inherited by nondegenerate in R^n (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we…
In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the…
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…
Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a…
We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes…
We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly…
Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto,…
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO…
We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $\alpha_1x_1+\alpha_2x_2 +y$ with {\it positive} integers $x_1,x_2$.
Wirsing's theorem on approximating algebraic numbers by algebraic numbers of bounded degree is a generalization of Roth's theorem in Diophantine approximation. We study variations of Wirsing's theorem where the inequality in the theorem is…
In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that…
We study the convergence of these operators in a weighted space of functions on a positive semi-axis and estimate the approximation by using a new type of weighted modulus of continuity and error estimation.
The Duffin-Schaeffer theorem is a well-known result from metric number theory, which generalises Khinchin's theorem from monotonic functions to a wider class of approximating functions. In recent years, there has been some interest in…
Khintchine's and Dyson's transference theorems can be very easily deduced from Mahler's transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem…