Related papers: Exact uniform approximation and Dirichlet spectrum…

We introduce an infinite family of approximations for a Dirichlet $L$-function $L(s, \chi)$ arising from truncated Euler products. These approximations are entire functions and satisfy the same functional equation as $L(s, \chi)$. We…

We prove that for any proper metric space $X$ and a function $\psi:(0,\infty)\to(0,\infty)$ from a suitable class of approximation functions, the Hausdorff dimensions of the set $W_\psi(Q)$ of all points $\psi$-well-approximable by a…

We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of…

In the Random Subset Sum Problem, given $n$ i.i.d. random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. Despite…

Let $f$ be a continuous monotone real function defined on a compact interval $[a,b]$ of the real line. Given a sequence of partitions of $[a,b]$, $% \Delta_n $, $\left\Vert {\Delta }_{n}\right\Vert \rightarrow 0$, and given $l\geq 0,m\geq…

Let $0<p\leq 1$, and let $\omega:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that…

In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement with arbitrary weight functions and establish a new and simple proof of this…

The spectrum of one-dimensional discrete Schr\"odinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly.…

We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\partial_t u_i}-\Delta u_i =\sum_{j=1}^{m}\beta_{ij} u_i^ru_j^{r+1}, \quad i=1,2,...,m$$ in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very recently,…

In this paper, we investigate Voiculescu's theorem on approximate unitary equivalence in separable properly infinite factors. As applications, we establish the norm-denseness of the set of all reducible operators, prove a generalized…

We show that for any $\epsilon<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions).…

We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems…

On the space $\mathcal{L}_{n+1}$ of unimodular lattices in $\mathbb{R}^{n+1}$, we consider the standard action of $a(t)=\mathrm{diag}(t^n,t^{-1},\ldots,t^{-1})\in \mathrm{SL}(n+1,\mathbb{R})$ for $t>1$. Let $M$ be a nondegenerate…

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to either Dirichlet or von Neumann boundary conditions and study both time-independent and time-periodic cases involving Liouville integrable…

We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences…

Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence.…

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $\Omega\subset\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of…

We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit…

In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem \[\begin{cases} -{\rm…

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(\lambda_t)_{t \in [0,1]}$ of invariant extensions of the linear Lebesgue measure $\lambda$ in…