Approximation to real numbers by cubic algebraic integers I

We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.

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 solution in integers x_0, x_1 and x_2 not all zero, and let omega(xi) denote the supremum of all real numbers omega such that, for each sufficiently large X, the dual inequalities |x_0+x_1*xi+x_2*xi^2| < X^{-omega}, |x_1| < X and |x_2| < X admit...

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|, \|q \zeta\| \} > q^{-1 + \tau}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.

We give some comments on W.M. Schmidt's theorem on Diophantine approximations with positive integers and our recent results on the topic.

We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general it implies new estimates on the least possible distance between conjugate algebraic numbers, which imp...

We introduce a new exponent of simultaneous rational approximation $\widehat{\lambda}_{\min}(\xi,\eta)$ for pairs of real numbers $\xi,\eta$, in complement to the classical exponents $\lambda(\xi,\eta)$ of best approximation, and $\widehat{\lambda}(\xi,\eta)$ of uniform approximation. It generalizes Fischler's exponent $\beta_0(\xi)$ in the sense that $\widehat{\lambda}_{\min}(\xi,\xi^2) = 1/\beta_0(\xi)$ whenever $\lambda(\xi,\xi^2) = 1$. Using parametric geometry of numbers...

The papers shows an algorithm to search for approximations of reals to rationals of the form a/b^2 that runs on \sqrt(b) polynomial time steps.

We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator.

Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary triangles and quadrilaterals by those with rational sides,diagonals and areas. We transform these problems into questions on the existence of infinitely many rational solutions on a two parameter family of quartic curves. This is further transf...

In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naive height tends to infinity. Let $I \subset \mathbb{R}$ be an arbitrary bounded interval, and $Q$ be a sufficiently large number. We obtain an asymptotic formula for the count of algebraic integers $\alpha$ of fixed degree $n$ and naive height $H(\alpha)\le Q$ lying in $I$. In this formula, we estimate the order of the error term from above and below. We show that algebra...