Mathematically Rectangular coordinate system The points whose horizontal and vertical coordinates are integers are called lattice points or whole points. 1. The area of the grid polygon must be integer Or half integer（ Odd number Half of). 2. The symmetric point of the grid point about the grid point is the grid point.
3. Area formula of grid polygon [1] , suppose that there are a grid point inside a grid point polygon and b grid points on the edge of the grid point polygon, and the area of the grid point polygon is S, then Pike formula There is S=a+b/2-1. 4. A lattice regular polygon can only be a square.
5. Grid point There is no other grid point on the triangle boundary, and there is a grid point inside, then the point is the triangle's focus 。 Lattice problem It is to study the number of lattice points in some special regions or even general regions. Problem on lattice point is also called integral point problem
The lattice problem originates from the study of the following two problems:
① Dirichlet Divisor problem, that is, finding D when x>1 two (x) =the number of lattice points in the region {1 ≤ u ≤ x, 1 ≤ v ≤ x, uv ≤ x}. In 1849, Dirichlet proved that D two (x) =xlnx+(2 ν - 1) x+△ (x), where ν is Euler constant ，△（x）=O（x zero point five ）。 The purpose of this problem is to make sure that the residual estimation △ (x)=O (x) is valid Infimum θ zero 。 ② For the lattice point problem in a circle, let x>1, A two (x) =number of lattice points in the circle μ+ν≤ x.
Gaussian Proved A two (x) =π x+R (x), where R (x)=O (x ^ 1/2), the problem of finding the infimum α of λ that makes the remainder estimate R (x)=O (x) true is called the lattice point problem in the circle or the Gaussian circle problem.
In 1903, Г. Ф. Voronoy proved that θ≤ 1/3;
In 1906, Shelpinsky proved that α ≤ 1/3;
In the 1930s, J.G. Copt proved that α ≤ 37/112, θ≤ 27/82;
In 1934-1935, E.C. Titchmarsh proved that α ≤ 15/46;
In 1963, Chen Jingrun Yin Wenlin proved that α ≤ 12/37; In 1950, Chi Zongtao proved that θ ≤ 15/46;
In 1953, H. Richter proved the same result;
In 1963, Yin Wenlin proved that θ ≤ 12/37;
In 1985, Г. A. Kolesnik proved that θ≤ 139/429;
In 1985, W.G. Nowak proved that α ≤ 139/429.
In terms of the lower limit,
In 1916, Hardy It is proved that α ≥ 1/4; In 1940, A.E. Ingham proved that θ ≥ 1/4.
People also speculated that θ=α=1/4, but still failed to prove it.
From this, the k-dimensional divisor problem, the lattice problem in the sphere and the lattice problem in the k-dimensional ellipsoid are directly generalized.
The knowledge points involved in lattice problems are usually Drawer principle Combined with graph theory knowledge, it is generally closely related to the parity and divisibility of integers. (1) In any 4n-3 integral points on the plane, n integral points must be taken so that their center of gravity is still an integral point.
(2) In 1983, Kemmitz conjecture, it was impossible to solve this difficult conjecture with elementary methods.
(3) Used in 2000 algebraic method It is successfully proved that the conjecture is correct when 4n-3 is replaced by 4n-2. (4) In 2003, German Reiher (born April 191984) unexpectedly combined algebraic methods with Combination method By ingenious combination, we captured the 20 year old Kemnitz conjecture. [2] 
