Quadratic approximation in ℚp

2014 ◽  
Vol 11 (01) ◽  
pp. 193-209 ◽  
Author(s):  
Yann Bugeaud ◽  
Tomislav Pejković

Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

2012 ◽  
Vol 148 (3) ◽  
pp. 718-750 ◽  
Author(s):  
Yann Bugeaud

AbstractWe establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.


1997 ◽  
Vol 122 (2) ◽  
pp. 193-205 ◽  
Author(s):  
HONGZE LI

In 1982, Glyn Harman [2] proved that for almost all n, the interval [n, n+n(1/10)+ε] contains a prime number. By this we mean that the set of n[les ]N for which the interval does not contain a prime has measure o(N) as n→+∞. It follows from Huxley's work [6] that if θ>1/6 then there will almost always be asymptotically nθ(log n)−1 primes in the interval [n, n+nθ]. In 1983, Glyn Harman [3] pointed that for almost all n, the interval [n, n+n(1/12)+ε] contains a prime number, and meantime Heath-Brown gave the outline of this result in [5]. The exponent was reduced to 1/13 by Jia [10], 2/27 by Li [12] and 1/14 by Jia [11], and meantime N. Watt [16] got the same result. In this paper we shall prove the following result.THEOREM. For almost all n, the intervalformula herecontains a prime number.


Author(s):  
D. E. Taylor ◽  
Ming-Yao Xu

AbstractGiven an infinite family of finite primitive groups, conditions are found which ensure that almost all the orbitals are not self-paired. If p is a prime number congruent to ±1(mod 10), these conditions apply to the groups P S L (2, p) acting on the cosets of a subgroup isomorphic to A5. In this way, infinitely many vertex-primitive ½-transitive graphs which are not metacirculants are obtained.


Author(s):  
L. D. PUSTYL'NIKOV

A new theory of generalized continued fractions for infinite-dimensional vectors with integer components is constructed. The results of this theory are applied to the classical problem on the distribution of quadratic residues and non-residues modulo a prime number and are based on the study of ergodic properties of some infinite-dimensional transformations.


1857 ◽  
Vol 3 ◽  
pp. 451-452
Author(s):  
Edward Sang

Mr Sang briefly explained an application of the method of continued fractions to the resolution of the exponential equation, and illustrated it by exhibiting the computation of the logarithm of the prime number 27073 directly.


2019 ◽  
Vol 155 (11) ◽  
pp. 2214-2233 ◽  
Author(s):  
Sam Chow ◽  
Niclas Technau

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.


Author(s):  
I. J. Good

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.


2016 ◽  
Vol 37 (4) ◽  
pp. 1323-1344
Author(s):  
JUN WU ◽  
JIAN-SHENG XIE

Let $\unicode[STIX]{x1D714}=[a_{1},a_{2},\ldots ]$ be the infinite expansion of a continued fraction for an irrational number $\unicode[STIX]{x1D714}\in (0,1)$, and let $R_{n}(\unicode[STIX]{x1D714})$ (respectively, $R_{n,k}(\unicode[STIX]{x1D714})$, $R_{n,k+}(\unicode[STIX]{x1D714})$) be the number of distinct partial quotients, each of which appears at least once (respectively, exactly $k$ times, at least $k$ times) in the sequence $a_{1},\ldots ,a_{n}$. In this paper, it is proved that, for Lebesgue almost all $\unicode[STIX]{x1D714}\in (0,1)$ and all $k\geq 1$, $$\begin{eqnarray}\displaystyle \lim _{n\rightarrow \infty }\frac{R_{n}(\unicode[STIX]{x1D714})}{\sqrt{n}}=\sqrt{\frac{\unicode[STIX]{x1D70B}}{\log 2}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n}(\unicode[STIX]{x1D714})}=\frac{C_{2k}^{k}}{(2k-1)\cdot 4^{k}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n,k+}(\unicode[STIX]{x1D714})}=\frac{1}{2k}.\end{eqnarray}$$ The Hausdorff dimensions of certain level sets about $R_{n}$ are discussed.


2008 ◽  
Vol 103 (1) ◽  
pp. 5
Author(s):  
Filip Saidak

The key result needed in almost all elementary proofs of the Prime Number Theorem is a prime number lemma proved by Atle Selberg in 1948. Without restricting ourselves to purely elementary techniques we show how the error term in Selberg's fundamental lemma relates to the error term in the Prime Number Theorem. In spite of all the interest in this topic over the last sixty years this particular question seems to have been overlooked in the past.


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