Related papers: Markoff-Lagrange spectrum and extremal numbers

Let xi be a real number which is neither rational nor quadratic over Q. Based on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any real number theta, there exist a constant c>0 and infinitely many non-zero…

For any irrational real number xi, let lambda(xi) denote the supremum of all real numbers lambda such that, for each sufficiently large X, the inequalities |x_0| < X, |x_0*xi-x_1| < X^{-lambda} and |x_0*xi^2-x_2| < X^{-lambda} admit a…

We call a positive real number $\lambda$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $\alpha$ such that $\mu(\alpha)=\lambda$. Here $\mu(\alpha)$ denotes the Lagrange constant of $\alpha$ -…

Let $\alpha$ and $\beta$ be real numbers such that $1$, $\alpha$ and $\beta$ are linearly independent over $\mathbb{Q}$. A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that…

Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(\Sigma,g)$. Precisely, if $\lambda_1(\Sigma)$ is the first eigenvalue of the…

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let…

The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed,…

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set \[ {\mathcal…

We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials,…

Let $w_{\alpha}(t)=t^{\alpha}\,e^{-t}$, $\alpha>-1$, be the Laguerre weight function, and $|\cdot|_{w_\alpha}$ denote the associated $L_2$-norm, i.e., $$ | f|_{w_\alpha}:=\Big(\int_{0}^{\infty}w_{\alpha}(t)| f(t)|^2\,dt\Big)^{1/2}. $$…

For a given irrational number $\alpha$ and a real number $\gamma$ in $(0,1)$ one defines the two-sided inhomogeneous approximation constant \begin{equation*} M(\alpha,\gamma):=\liminf_{|n|\rightarrow\infty}|n| ||n\alpha-\gamma||,…

A symmetrized lattice of $2n$ points in terms of an irrational real number $\alpha$ is considered in the unit square, as in the theorem of Davenport. If $\alpha$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be…

Let $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, be the Gegenbauer weight function, and $\Vert\cdot\Vert$ denote the associated $L_2$-norm, i.e., $$ \Vert f\Vert:=\Big(\int_{-1}^{1}w_{\lambda}(t)\vert f(t)\vert^2\,dt\Big)^{1/2}.…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…

For the Gegenbauer weight function $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, we denote by $\Vert\cdot\Vert_{w_{\lambda}}$ the associated $L_2$-norm, $$ \Vert…

We prove that if $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are non-zero real numbers, not all of the same sign, $\lambda_1 / \lambda_2$ is irrational, and $\varpi$ is any real number then, for any $\eps > 0$ the inequality $…

Given a conic $\mathcal{C}$ over $\mathbb{Q}$, it is natural to ask what real points on $\mathcal{C}$ are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least…

We consider Lagrange's equation $x_1^2 + x_2^2 + x_3^2 + x_4^2 = N$, where $N$ is a sufficiently large and odd integer, and prove that it has a solution in natural numbers $x_1, \dots, x_4 $ such that $x_1 x_2 x_3 x_4 + 1$ has no more than…

The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point $4/\sqrt3$. We show that both the Markoff…