scholarly journals On a Counting Theorem for Weakly Admissible Lattices

2020 ◽  
Author(s):  
Reynold Fregoli
Keyword(s):  
Diophantine Approximation ◽  
General Setting ◽  
Upper Bounds ◽  
Lattice Points ◽  
Precise Estimate ◽  
Linear Subspaces ◽  
Minimal Structure ◽  
Partition Method

Abstract We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve “hyperbolic spikes” and occur naturally in multiplicative Diophantine approximation. We use Wilkie’s o-minimal structure $\mathbb{R}_{\exp }$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The 1st one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The 2nd application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result, we develop a sophisticated partition method that is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

Equidistribution of Farey sequences on horospheres in covers of and applications

2019 ◽  
Vol 41 (2) ◽  
pp. 471-493
Author(s):  
BYRON HEERSINK
Keyword(s):  
Diophantine Approximation ◽  
Lattice Points ◽  
Index Subgroup ◽  
Farey Sequence ◽  
Frobenius Numbers ◽  
Farey Sequences ◽  
Approximation Problems ◽  
Primitive Lattice Points ◽  
Finite Index Subgroup ◽  
Statistical Results

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

scholarly journals Lifting, restricting and sifting integral points on affine homogeneous varieties

2012 ◽  
Vol 148 (6) ◽  
pp. 1695-1716 ◽  
Author(s):  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Homogeneous Spaces ◽  
Acta Math ◽  
Upper Bounds ◽  
Lattice Points ◽  
List Type ◽  
Integral Points ◽  
Counting Problem ◽  
Symmetric Varieties ◽  
Almost Prime ◽  
Homogeneous Varieties

AbstractIn [Gorodnik and Nevo,Counting lattice points, J. Reine Angew. Math.663(2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak,Prime and almost prime integral points on principal homogeneous spaces, Acta Math.205(2010), 361–402] and use them to establish several useful consequences of this property, including:(1)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

Estimates of lattice points in the discriminant aspect over abelian extension fields

2018 ◽  
Vol 30 (3) ◽  
pp. 767-773 ◽  
Author(s):  
Wataru Takeda ◽  
Shin-ya Koyama
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Abelian Extension ◽  
Lattice Points ◽  
Abelian Extensions ◽  
Lindelöf Hypothesis ◽  
Error Terms

AbstractWe estimate the number of relatively r-prime lattice points in {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

scholarly journals TORSIONAL RIGIDITY OF MINIMAL SUBMANIFOLDS

2006 ◽  
Vol 93 (1) ◽  
pp. 253-272 ◽  
Author(s):  
STEEN MARKVORSEN ◽  
VICENTE PALMER
Keyword(s):  
Warped Product ◽  
Torsional Rigidity ◽  
Exit Time ◽  
General Setting ◽  
Minimal Submanifolds ◽  
Upper Bounds ◽  
Product Model ◽  
Product Comparison ◽  
Mean Exit Time ◽  
Model Spaces

We prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds $P^m$ in ambient Riemannian manifolds $N^n$ with a pole $p$. The upper bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained via previously established isoperimetric inequalities, which are here extended to hold for this more general setting based on warped product comparison spaces. We also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for Brownian motion at infinity is finite.

The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

2019 ◽  
Vol 18 (11) ◽  
pp. 1950212 ◽  
Author(s):  
Honglin Zou ◽  
Dijana Mosić ◽  
Jianlong Chen
Keyword(s):  
Sufficient Conditions ◽  
General Setting ◽  
Upper Bounds ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Operator Matrices ◽  
Skew Field ◽  
Block Matrices ◽  
Necessary And Sufficient ◽  
Drazin Index

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

scholarly journals Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

2020 ◽  
Vol 2020 (767) ◽  
pp. 77-107 ◽  
Author(s):  
Aaron Levin ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Diophantine Approximation ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Counting Function ◽  
Upper Bounds ◽  
General Version ◽  
Moving Targets ◽  
Algebraic Tori ◽  
Greatest Common Divisors ◽  
Common Zeros

AbstractWe study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural “asymptotic gcd” inequality of Pasten and the second author, and consider moving targets versions of our results.

Cushions, cigars and diamonds: an area-perimeter problem for symmetric ovals

1979 ◽  
Vol 85 (1) ◽  
pp. 1-16 ◽  
Author(s):  
H. T. Croft
Keyword(s):  
Upper Bound ◽  
Natural Condition ◽  
Lattice Point ◽  
Convex Set ◽  
General Setting ◽  
Lattice Points ◽  
Two Dimensional ◽  
Algebraic Expressions ◽  
Image Position ◽  
Closed Convex Set

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximizewhere κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.

scholarly journals Doubling coverings via resolution of singularities and preparation

2020 ◽  
pp. 2050018
Author(s):  
Raf Cluckers ◽  
Omer Friedland ◽  
Yosef Yomdin
Keyword(s):  
Level Sets ◽  
Arbitrary Dimension ◽  
General Setting ◽  
Upper Bounds ◽  
Algebraic Set ◽  
Dimension Formula ◽  
First Case ◽  
Polynomial Zero ◽  
Subanalytic Sets ◽  
Low Dimensional

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form [Formula: see text] This is done in a rather general setting, i.e. for the [Formula: see text]-complement of a polynomial zero-level hypersurface [Formula: see text] and for the regular level hypersurfaces [Formula: see text] themselves with no assumptions on the singularities of [Formula: see text]. The coefficient [Formula: see text] is the ambient dimension [Formula: see text] in the first case and [Formula: see text] in the second case. However, the question of a uniform behavior of the coefficient [Formula: see text] remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set [Formula: see text] of dimension [Formula: see text] away from the [Formula: see text]-neighborhood of a lower dimensional set [Formula: see text], with bound of the form [Formula: see text] holding uniformly in the complexity of [Formula: see text]. We also show an analogue for level sets with parameter away from the [Formula: see text]-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

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