Distribution of Algebraic Numbers and Metric Theory of Diophantine Approximation
In this paper we give an overview of recent results regarding close conjugate algebraic numbers, the number of integral polynomials with small discriminant and pairs of polynomials with small resultants.
- PDF / 319,731 Bytes
- 26 Pages / 439.36 x 666.15 pts Page_size
- 74 Downloads / 266 Views
Abstract In this paper we give an overview of recent results regarding close conjugate algebraic numbers, the number of integral polynomials with small discriminant and pairs of polynomials with small resultants. Keywords Diophantine approximation • approximation by algebraic numbers • discriminant • resultant • polynomial root separation 2010 Mathematics Subject Classification. 11J83, 11J13, 11K60, 11K55
1 Introduction Throughout the paper A stands for the Lebesgue measure of a measurable set A R and dim B denotes the Hausdorff dimension of B. Given W N ! .0 C1/, let L. / denote the set of x 2 R such that ˇ ˇ ˇ ˇ ˇx p ˇ < ˇ qˇ
.q/ q
(1)
has infinitely many solutions .p; q/ 2 Z N. We begin by recalling two classical results in metric theory of Diophantine approximation.
V. Bernik O. Kukso Institute of Mathematics, Academy of Sciences of Belarus, Minsk, Belarus V. Beresnevich () Department of Mathematics, University of York, Heslington, York, England F. G¨otze Faculty of Mathematics, Bielefeld University, Bielefeld, Germany P. Eichelsbacher et al. (eds.), Limit Theorems in Probability, Statistics and Number Theory, Springer Proceedings in Mathematics & Statistics 42, DOI 10.1007/978-3-642-36068-8 2, © Springer-Verlag Berlin Heidelberg 2013
23
24
V. Bernik et al.
W N ! .0; C1/ be monotonic and I be an
Khintchine’s theorem [35]. Let interval in R. Then .I \ L. // D
8 < 0;
if
P1
:.I /;
if
P1
qD1
.q/ < 1;
qD1
.q/ D 1:
Jarn´ık–Besicovitch theorem [25, 34]. Let v > 1 and for q 2 N let Then 2 : dim L. v / D vC1
(2)
v .q/
D q v .
The condition that is monotonic can be omitted from the convergence case of Khintchine’s theorem, though it is vital in the case of divergence—see [12, 33, 42] for a further discussion. By the turn of the millennium the above theorems were generalised in various directions. One important direction of research has been Diophantine approximation by algebraic numbers and/or integral polynomials, which has eventually grown into an area of number theory known as Diophantine approximation on manifolds. Given a polynomial P D an x n C C a1 x C a0 2 ZŒx, the number H D H.P / D max0i n jaj j will be called the (naive) height of P . Given n 2 N and an approximation function ‰ W N ! .0; C1/, let Ln .‰/ be the set of x 2 R such that jP .x/j < ‰.H.P //
(3)
for infinitely many P 2 ZŒx n f0g with deg P n. Note that L1 .‰/ is essentially the same as the set L.‰/ introduced above. Thus, the following statement represents an analogue of Khintchine’s theorem for the case of polynomials. Theorem 1. Let n 2 N and ‰ W N ! .0; C1/ be monotonic. Then for any interval I .I \ Ln .‰// D
8 < 0;
if
P1
:.I /;
if
P1
hD1
hn1 ‰.h/ < 1;
n1 ‰.h/ D 1: hD1 h
(4)
The case of convergence of Theorem 1 was proved in [17], the case of divergence was proved in [4]. The condition that ‰ is monotonic can be omitted from the case of convergence as shown in [6]. Theorem 1 was generalised to the case of approximation in the fields of complex and p-adic numbers [9, 19], to simulta
Data Loading...
