Related papers: Some remarks on Mahler's classification in higher …
By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by…
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…
We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…
We develop a theory of diophantine approximation on generalized flag varieties, varieties that can be obtained as a quotient of a semisimple algebraic group by a parabolic subgroup. Using methods from the theory of arithmetic groups, due in…
We establish Diophantine type estimates on shifts of trigonometric polynomials on the torus $\mathbb{T}^d$, as well as that of their square roots. These estimates arise from the spectral analysis of the quasi-periodic Schr\"odinger and the…
We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions…
This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a…
We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big…
We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between…
We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric…
In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $\alpha$ is an irrational number having a continued…
We establish Diophantine inequalities for the fractional parts of generalized polynomials $f$, in particular for sequences $\nu(n)=\lfloor n^c\rfloor+n^k$ with $c>1$ a non-integral real number and $k\in\mathbb{N}$, as well as for $\nu(p)$…
The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on non-degenerate manifolds, which completes earlier results for convergence.
We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower…
Let $T_\beta$ be the $\beta$-transformation on $[0,1)$ defined by $$T_\beta(x)=\beta x\text{ mod }1.$$ We study the Diophantine approximation of the orbit of a point $x$ under $T_\beta$. Precisely, for given two positive functions…
A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models…
This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral. For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
This paper settles recent conjectures concerning the $p$-adic Haar measure applied to a family of sets defined in terms of Diophantine approximation. This is done by determining the spectrum of measure values for each family and seeing that…
Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support {\tau}-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given…