Related papers: On distribution of fractional parts of linear form…

We present a proof of a multidimensional version of Peres-Schlag's theorem on Diophantine approximations with lacunary sequences.

We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood…

We prove a discrepancy estimate related to the sequence of fractional parts of $b^n/n$. This improves an earlier result of Cilleruelo et al.

We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…

We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of…

We study the well-posedness of a semilinear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map.…

We give a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution. We also obtain an explicit description for the set of all bounded (or decay)…

This paper concerns extension of the classical Lagrange theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this…

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…

We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.

We consider the continued fraction expansion of real numbers under the action of a non-uniform lattice in PSL(2,R) and prove metric relations between the convergents and a natural geometric notion of good approximations.

Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…

In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and…

This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. The emphasis is on the interplay between regularity,…

Assume that $\alpha>1$ is an algebraic number and $\xi\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\xi \alpha^{n})$. Under various Diophantine conditions on $\xi$ and $\alpha$, we…

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…

In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.

We use a concept of weak asymptotic solution for homogeneous as well as non-homogeneous fractional advection dispersion type equations. Using Legendre scaling functions as basis, a numerical method based on Galerkin approximation is…

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of…