Diophantine Approximation with Two Primes and One Square of Prime

We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + λ2 p2 + λ3 p2 3 + η| < (max pj )- 1/40 (log max pj ) 4 .

A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION

Let Q be an infinite set of positive integers, τ 〉 1 be a real number and let Wτ（Q）={x∈R：｜x-p/q｜〈^-τ for infinitely many （p,q）∈ Z×Q}.For any given positive integer m, set Q（m）={n∈N：（n,m）=1}. If m is divisible by at least two prime factors, Adiceam [1] showed that Wτ（N） ／ Wτ（Q（m）） contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers？ In this note, we give an affirmative answer to Adiceam＇s question.

VALUE DISTRIBUTION THEORY AND DIOPHANTINE APPROXIMATION

In this paper, we will introduce some problems and results between Diophantine approximation and value distribution theory.

一类Brahmagupta-Fermat-Pell方程x^2-dy^2=±1(英文)

This paper corresponds to the written versions of many lectures at several locations including the most recent one at Weinan Teachers University on June 8,2011.I would like to thank Professor Hailong Li for inviting me to publish this in the journal of his university.I wish also to express my deep gratitude to my friend Shigeru Kanemitsu,thanks to whom I could visit Weinan Teachers University,and who also came up with a written version of these notes. The topic is centered around the equation x2-dy2=±1,which is important because it produces the(infinitely many) units of real quadratic fields.This equation,where the unknowns x and y are positive integers while d is a fixed positive integer which is not a square,has been mistakenly called with the name of Pell by Euler.It was investigated by Indian mathematicians since Brahmagupta(628) who solved the case d=92,then by Bhaskara II(1150) for d=61 and Narayana(during the 14-th Century) for d=103.The smallest solution of x2-dy2=1 for these values of d are respectively 1 1512-92·1202=1, 1 766 319 0492-61·226 153 9802=1 and 227 5282-103·22 4192=1, and hence they could not have been found by a brute force search! After a short introduction to this long story of Pell's equation,we explain its connection with Diophantine approximation and continued fractions(which have close connection with the structure of real quadratic fields),and we conclude by saying a few words on more recent developments of the subject in terms of varieties.Finally we mention applications of continued fraction expansion to electrical circuits.

SYMMETRIC AND ASYMMETRIC DIOPHANTINE APPROXIMATION

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].

Sharp Bounds for Symmetric and Asymmetric Diophantine Approximation

In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results with a geometric method and give both sharp upper and lower bounds.The asymptotic frequencies that these bounds occur are also calculated.

The Degenerated Second Main Theorem and Schmidt's Subspace Theorem

In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.

On the Number of Polynomials with Small Discriminants in the Euclidean and p-adic Metrics

In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy certain polynomial and derivative conditions is also determined.