Approximation to real numbers by cubic algebraic integers I

In this paper, we present a result on using algebraic conjugates to form a sequence of approximations to an algebraic number, and in this way obtain effective irrationality measures for related algebraic numbers. From this result, we are able to generalise Thue's Fundamentaltheorem.

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.

In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their na\"{\i}ve height tends to infinity. For an arbitrary interval $I \subset \mathbb{R}$ and sufficiently large $Q>0$, we obtain an asymptotic formula for the number of algebraic integers $\alpha\in I$ of fixed degree $n$ and na\"{\i}ve height $H(\alpha)\le Q$. In particular, we show that the real algebraic integers of degree $n$, with their height growing, tend to be distribut...

The original version of this paper did not take into account that there may be solutions (x_0, y_o)in Z X Z of f(x,y) = x^3 + p(y)x + q(y) = 0 even though w_0 = (-3D(y_0))^(1/2) is irrational.

The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.

We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms.

In this paper we consider the problem of counting algebraic numbers $\alpha$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_{\alpha}(x)$ of $\alpha$ is bounded, $|P_{\alpha}'(\alpha)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $\alpha$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\fr...

Let $\alpha$ be a Pisot number. Let $L(\alpha)$ be the largest positive number such that for some $\xi=\xi(\alpha)\in \mathbb R$ the limit points of the sequence of fractional parts $\{\xi \alpha^n\}_{n=1}^{\infty}$ all lie in the interval $[L(\alpha), 1-L(\alpha)]$. In this paper we show that if $\alpha$ is of degree at most 4 or $\alpha\le \frac{\sqrt 5 + 1}{2}$ then $L(\alpha)\ge \frac{3}{17}$. Also we find explicitly the value of $L(\alpha)$ for certain Pisot numbers of d...

Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height \hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points, \hat{h}_f is bounded below by a positive constant (depending only on d) times some ki...

Let $n \ge 2$ be an integer and $\xi$ a transcendental real number. We establish several new relations between the values at $\xi$ of the exponents of Diophantine approximation $w_n, w_{n}^{\ast}, \hat{w}_{n}$, and $\hat{w}_{n}^{\ast}$. Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality $\hat{w}_{n}(\xi) \le 2n-1$ proved by Davenport and Schmidt in 1969.