Approximation to real numbers by cubic algebraic integers I

It has been established on many occasions that the set of quotients of prime numbers is dense in the set of positive real numbers. More recently, it has been proved that the set of quotients of primes in the Gaussian integers is dense in the complex plane. In this article, we not only extend this result to any imaginary quadratic number ring, but also prove that the set of quotients of primes in any real quadratic number ring is dense in the set of real numbers. To conclude, ...

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-\nu}$. The latest record in this regard is Kaisa Matom\"aki's landmark result $\nu=1/3-\varepsilon$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Ma...

W.M.Schmit[11] conjectured that for any$\;\theta$ with deg$\;\theta\geq 3,$ there is no constant$\;C=C(\theta)$ so that$\;|p-q\theta|>Cq^{-1}$ for every rationa$\;p/q.$ [12,p26] states that the computations of the first several thousand partial quotients for such numbers as$\;\sqrt[3]{2}$ and$\;\sqrt[3]{3}$ support the conjecture that the sequence of partial quotients is unbounded. In this paper, applying Dirichlet's approximation theorem to certain algebraic numbers$\;\the...

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for $\theta(k,l;x)$ and $\psi(k,l;x)$ for $k=1,3,4,6...

Given a non-singular diagonal cubic hypersurface $X\subset\mathbb{P}^{n-1}$ over $\mathbb{F}_q(t)$ with $\mathrm{char} (\mathbb{F}_q)\neq 3$, we show that the number of rational points of height at most $|P|$ is $O(|P|^{3+\varepsilon})$ for $n=6$ and $O(\lvert P \rvert^{2+\varepsilon})$ for $n=4$. In fact, if $n=4$ and $\mathrm{char}(\mathbb{F}_q) >3$ we prove that the number of rational points away from any rational line contained in $X$ is bounded by $O(|P|^{3/2+\varepsilon...

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be related to the congruence equation problem $x y \equiv c \pmod q$ with $1 \leq x, y \leq q^{1/2 + \epsilon}$.

We study rational approximation properties for successive powers of extremal numbers defined by Roy. For $n\in{\{1,2\}}$, the classic approximation constants $\lambda_{n}(\zeta),\hat{\lambda}_{n}(\zeta),w_{n}(\zeta),\hat{w}_{n}(\zeta)$ connected to an extremal number $\zeta$ have been established and in fact much more is known. However, so far almost nothing had been known for $n\geq 3$. In this paper we determine all classic approximation constants as above for $n=3$. Our me...

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $\xi$ can be approximated by algebraic numbers $\alpha$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(\alpha)$ of $\alpha$. He showed that the infimum $\omega^*_n(\xi)$ of all $\omega$ for which infinitely many such $\alpha$ have $|\xi-\alpha| \le H(\alpha)^{-\omega-1}$ is at least $(n+1)/2$. He also asked if we could even have $\omega^*_n...

We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat cubic, we find all integer solutions of |x^3+y^3-z^3|<M with 0<x<=y<z<N in heuristic time << L(N) M [where L(X):= (log X)^O(1)] provided M>>N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M<N^3), the computatio...

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\geq 2$) of $\zeta_{1},\zeta_{2},...,\zeta_{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$ with prescribed approximation pr...