scholarly journals Lattice Simplices with a Fixed Positive Number of Interior Lattice Points: A Nearly Optimal Volume Bound

2018 ◽  
Vol 2020 (13) ◽  
pp. 3871-3885
Author(s):  
Gennadiy Averkov ◽  
Jan Krümpelmann ◽  
Benjamin Nill
Keyword(s):  
Upper Bound ◽  
Lattice Points ◽  
Linear Factor ◽  
Fixed Positive Number

Abstract We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves significantly upon the previously best results by Pikhurko from 2001.

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 Circles on lattices and maximum determinant matrices

2020 ◽  
pp. 2-11
Author(s):  
Nikolay Balonin ◽  
Mikhail Sergeev ◽  
Jennifer Seberry ◽  
Olga Sinitshina
Keyword(s):  
Upper Bound ◽  
Lattice Points ◽  
Video Data ◽  
Practical Significance ◽  
Circulant Matrices ◽  
Special Role ◽  
Maximum Order ◽  
Fermat Numbers ◽  
Fixed Structure ◽  
Special Order

Introduction: The Hadamard conjecture about the existence of maximum determinant matrices in all orders multiple of 4 is closely related to Gauss's problem about the number of points with integer coordinates (Z3 lattice points) on a spheroid, cone, paraboloid or parabola. The location of these points dictates the number and types of extreme matrices. Purpose: Finding out how Gaussian points on sections of solids of revolution are related to the number and types of maximum determinant matrices with a fixed structure for odd orders. Specifying a precise upper bound of maximum determinant values for edged two-circulant matrices and the orders on which they prevail over simpler cyclic structures. Results: A newly proposed formula refines the overly optimistic Elich – Wojtas’ upper bound for the case of matrices with а fixed structure. Fermat numbers have a special role for orders of 4t + 1, and Barba numbers affect the formation of classes of maximum determinant matrices which occupy the areas of orders 4t + 3, successively replacing each other. For a two-circulant structure with an edge, the maximum order of an optimal symmetric solution is estimated as 67. It is proved that the determinant of edged block matrices is superior to the determinants of circulant matrices everywhere except for a special order 39. Practical relevance: Maximum (for a fixed structure) determinant matrices related to lattice points have a direct practical significance for noise-resistant coding, compression and masking of video data.

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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