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1.
Glyn Harman 《Monatshefte für Mathematik》2010,137(2):51-57
Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that
|\fracs(n)n-a| < n-0.52.\left|\frac{\sigma(n)}{n}-\alpha \right| < n^{-0.52}. 相似文献
2.
Glyn Harman 《Monatshefte für Mathematik》2010,160(1):51-57
Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that $$\left|\frac{\sigma(n)}{n}-\alpha \right| < n^{-0.52}.$$ We prove this result by modifying an argument given by Wolke (Monatsh Math 83:163–166, 1977) which in its original form could not produce an exponent greater than 0.5. We also explain how the exponent can be improved to 0.61 on the Riemann Hypothesis. 相似文献
3.
Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for
naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type.
The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University
Research Fellow.
For Iona and Ayesha on No. 3. 相似文献
4.
Yuqing Zhang 《Journal of Number Theory》2009,129(8):1976-1989
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of Rn and their nondegenerate submanifolds. 相似文献
5.
6.
José Felipe Voloch 《Monatshefte für Mathematik》1995,119(4):321-325
We study diophantine approximations to algebraic functions in characteristicp. We improve a theorem of Osgood, and give two classes of examples showing that this result is nearly sharp. One of these classes exhibits a new phenomenon. 相似文献
7.
In this paper we study the set of x ∈ [0, 1] for which the inequality |x − x
n
| < z
n
holds for infinitely many n = 1, 2, .... Here x
n
∈ [0, 1) and z
n
s> 0, z
n
→ 0, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution
functions of sequences, we find the asymptotic density of n for which |x − x
n
| < z
n
, where x is a discontinuity point of some distribution function of x
n
. Generally, we also prove, for an arbitrary sequence x
n
, that there exists z
n
such that the density of n = 1, 2, ..., x
n
→ x, is the same as the density of n = 1, 2, ..., |x − x
n
| < z
n
, for x ∈ [0, 1]. Finally we prove, using the longest gap d
n
in the finite sequence x
1, x
2, ..., x
n
, that if d
n
≤ z
n
for all n, z
n
→ 0, and z
n
is non-increasing, then |x − x
n
| < z
n
holds for infinitely many n and for almost all x ∈ [0, 1]. 相似文献
8.
Zero-infinity laws in Diophantine approximation 总被引:1,自引:0,他引:1
9.
10.
Asmus L. Schmidt 《Journal of Number Theory》1983,16(2):169-204
A complete determination of the discrete part of the approximation spectrum and the Markoff spectrum is given for the Eisensteinian field. 相似文献
11.
Mariusz Urbański 《Journal of Number Theory》2005,110(2):219-235
It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions. 相似文献
12.
Yann Bugeaud 《Mathematische Annalen》2008,341(3):677-684
We provide an explicit construction of elements of the middle third Cantor set with any prescribed irrationality exponent.
This answers a question posed by Kurt Mahler. 相似文献
13.
We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems
and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed
maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times. 相似文献
14.
Damien Roy 《Monatshefte für Mathematik》2014,173(3):417-432
Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper. 相似文献
15.
Junjiro Noguchi 《中国科学A辑(英文版)》2005,48(Z1)
We discuss the analogue of the Nevanlinna theory and the theory of Diophan-tine approximation, focussing on the second main theorem and abc-conjecture. 相似文献
16.
17.
18.
Let (X,d) be a metric space and (Ω,d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups). 相似文献
19.
Emre Alkan 《Journal of Number Theory》2006,118(1):1-14
Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα, with n B-free. 相似文献
20.
Best simultaneous approximation of bounded functions with values in certain Banach spaces 总被引:1,自引:0,他引:1
Jaroslav Mach 《Mathematische Annalen》1979,240(2):157-164
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