scholarly journals Diophantine approximation with the constant right-hand side of inequalities on short intervals

2021 ◽  
Vol 65 (4) ◽  
pp. 397-403
Author(s):  
V. I. Bernik ◽  
D. V. Vasilyev ◽  
E. V. Zasimovich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Short Intervals ◽  
Right Hand ◽  
Best Estimate ◽  
Diophantine Approximations

In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | < Q-w, w > n, Q >1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate µ B< c1(n) Q – (w-1/n). The best estimate to date was c2(n) Q –(w- n/n).

scholarly journals Diophantine approximations with a constant right-hand side of inequalities on short intervals. 1

2021 ◽  
Vol 65 (5) ◽  
pp. 526-532
Author(s):  
V. I. Bernik ◽  
N. V. Budarina ◽  
E. V. Zasimovich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Short Intervals ◽  
Integer Polynomials ◽  
Right Hand ◽  
Diophantine Approximations

The problem of finding the Lebesgue measure 𝛍 of the set B1 of the coverings of the solutions of the inequality, ⎸Px⎹ <Q−w, w>n , Q ∈ N and Q >1, in integer polynomials P (x) of degree, which doesn’t exceed n and the height H (P) ≤ Q , is one of the main problems in the metric theory of the Diophantine approximation. We have obtained a new bound 𝛍B1 <c(n)Q−w+n, n<w<n+1, that is the most powerful to date. Even an ineffective version of this bound allowed V. G. Sprindzuk to solve Mahler’s famous problem.

scholarly journals A mass transference principle for systems of linear forms and its applications

2018 ◽  
Vol 154 (5) ◽  
pp. 1014-1047 ◽  
Author(s):  
Demi Allen ◽  
Victor Beresnevich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Linear Forms ◽  
Transference Principle ◽  
Integer Points ◽  
Additional Constraints

In this paper we establish a general form of the mass transference principle for systems of linear forms conjectured in 2009. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine–Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine–Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira.

scholarly journals On orbits of unipotent flows on homogeneous spaces, II

1986 ◽  
Vol 6 (2) ◽  
pp. 167-182 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Invariant Subspace ◽  
Lebesgue Measure ◽  
Compact Subset ◽  
Lie Group ◽  
Diophantine Approximation ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Semisimple Lie Group ◽  
Unipotent Subgroup ◽  
Unipotent Flows

AbstractWe show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

Some theorems in the metric theory of diophantine approximation

1986 ◽  
Vol 99 (3) ◽  
pp. 385-394 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Additional Constraint ◽  
Number Of Solutions ◽  
Initial Question ◽  
Image Position ◽  
Almost All

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequalityfor almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint on f, the number of solutions of (1) is asymptotically equal toas k tends to infinity.

scholarly journals Introduction to Diophantine Approximation. Part II

2017 ◽  
Vol 25 (4) ◽  
pp. 283-288
Author(s):  
Yasushige Watase
Keyword(s):  
Diophantine Approximation ◽  
Irrational Number ◽  
Basic Result ◽  
Formal Proof ◽  
Integer Solution ◽  
Linear Forms ◽  
Diophantine Approximations ◽  
Integer Solutions ◽  
System 1 ◽  
Hurwitz Theorem

SummaryIn the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2− a2b1| ≠ 0, has at least one integer solution.

A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM

2014 ◽  
Vol 91 (1) ◽  
pp. 34-40 ◽  
Author(s):  
YUEHUA GE ◽  
FAN LÜ
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Real Numbers ◽  
Image Position

AbstractWe study the distribution of the orbits of real numbers under the beta-transformation$T_{{\it\beta}}$for any${\it\beta}>1$. More precisely, for any real number${\it\beta}>1$and a positive function${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

Metric Diophantine approximation with two restricted variables I. Two square-free integers, or integers in arithmetic progressions

1988 ◽  
Vol 103 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Arithmetic Progressions ◽  
Integer Variable ◽  
Number Of Solutions ◽  
Metric Diophantine Approximation ◽  
Image Position ◽  
Almost All

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequalityfor almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sumdiverges, or converges, respectively.

scholarly journals The Non-homogeneous Groshev Convergence theorem for Diophantine Approximation on Manifolds

2017 ◽  
Vol 13 (4) ◽  
pp. 7354-7369
Author(s):  
Faiza Akram
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Convergence Theorem ◽  
Type Theory ◽  
Major Work ◽  
Extreme Measure

This paper is based on Khintchine theorem, Groshev theorem and measure and dimension theorems for non-degenerate manifolds. The inhomogeneous Diophantine approximation of Groshev type on manifolds is studied. Major work is to discuss the inhomogeneous convergent theory of Diophantine approximation restricted to non-degenerate manifold in , based on the proof of Barker-Sprindzuk conjecture, the homogeneous theory of Diophantine approximation and inhomogeneous Groshev type theory for Diophantine approximation, by the decomposition of the set in manifold, with the aid of Borel Cantell lemma and transformation of lemma and its properties and the main inhomogeneous conversion principle, we know these two types of set in sense of Lebesgue measure is zero provided that the convergent sum condition is satisfied, from which several conclusions about the inhomogeneous convergent theory of Diophantine approximation is obtained. The main result is that Lebesgue measure is inhomogeneous strongly extremal. At last we use the fact that friendly measure is strongly contracting measure to develop an inhomogeneous strong extreme measure which is restricted to matrices with dependent quantities

A PROBLEM OF ERDÖS, SZÖSZ AND TURÁN CONCERNING DIOPHANTINE APPROXIMATIONS

2008 ◽  
Vol 04 (04) ◽  
pp. 691-708 ◽  
Author(s):  
FLORIN P. BOCA
Keyword(s):  
Lebesgue Measure ◽  
Complete Solution ◽  
Probabilistic Argument ◽  
Diophantine Approximations ◽  
Coprime Integers

For A > 0 and c > 1, let S(N, A, c) denote the set of those numbers θ ∈ ]0,1[ which satisfy [Formula: see text] for some coprime integers a and b with N < b ≤ cN. The problem of the existence and computation of the limit f(A, c) of the Lebesgue measure of S(N, A, c) as N → ∞ was raised by Erdös, Szüsz and Turán [3]. This limit has been shown to exist by Kesten and Sós [5] using a probabilistic argument and explicitly computed when Ac ≤ 1 by Kesten [4]. We give a complete solution proving directly the existence of this limit and identifying it in all cases.

Diophantine approximation in Kleinian groups

1994 ◽  
Vol 116 (1) ◽  
pp. 57-78 ◽  
Author(s):  
Bernd Stratmann
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Detailed Understanding ◽  
Diophantine Approximations ◽  
Natural Way

AbstractThe δ-homogeneity of the Patterson measure is used for a closer study of the limit sets of Kleinian groups. A combination of the properties of this measure with concepts of diophantine approximations is shown to lead to a more detailed understanding of these limit sets. In particular, it is seen to how great an extent the studies of these sets, in terms of Hausdorff measure or Hausdorff dimension, are limited in a natural way.

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