（2School of Mathematics， Shandong University， JinanShandong250100China）
　　Foundation item： Supported by the National Natural Science Foundation of China and Ministry of Education in China
　　Biography： Hua GuoDong， male， professor in Shanghai Mathematical center， major in number theory and its application.
　　Abstract： In this paper， we investigate the property of the lines with irrational slope concerning the lattice points. We prove that the lines with irrational slope either passing through one lattice point or contains no lattice points. More precisely， we prove for those lines with irrational slope which do not passing through any lattice points， which have the property that infinite number of lattice points on either side of the the line that lying closer to the line at any assigned distance.
　　Key words： Lattice； Irrational slope lines； Bolzano-Weierstrass theorem
　　CLC Number： O156.3AMS（2000）Subject Classification： 11H06； 51M05Document code： A 【文章編号】2236-1879（2018）07-0036-03
　　1. Introduction
　　The geometry of numbers is a very interesting discipline that it relates number theory with geometry， and its makes number theory from the viewpoint be visual and concrete. The branch of geometry of numbers is fundamentally built by Hermann Minkowski， a German mathematician makes the subject a independent branch in number theory. Hermann Minkowski has contributed a lot in mathematics， such as functional analysis， algebraic number theory， mathematical physics， etc. Hermann Minkowski belongs to the mathematical legendary person that makes geometry of numbers a very beautiful and elegant subject that has some deep results and wide applications in mathematics. The fundamental concept in the geometry of numbers is lattice point， its defined as the points with integer coordinates （x，y） ， where x，y∈Z . In this paper， we use the definition of lattice points to be those points which coordinate （x，y） satisfy x，y∈Z， furthermore it has no other restriction. The very famous fundamental theorem of Minkowski state as follows：
　　Let R be a convex body in n-dimensional space which is symmetrical about the origin and has a volume vol（S）>2n. Then R must contains a no-zero lattice point.
　　This theorem is deep and very elegant that it reveals deep connection between the volume of the convex body and the lattice points. This deep theorem which would not be proved in this paper， since its not the focus of this paper.
　　Many other famous mathematicians has also contributed to the geometry of numbers， the most famous mathematicians could be Gauss and Dirichlet. Gauss studied the circle problem， if we denote the lattice points inside the circle x2+y2R to beC（R） . Then Gauss gives the following affirmative answer： 　　C（R）=πR+O（R） （1.1）
　　where O（R） denote the function f（x） satisfy f（x）c（R） ， and here R should be a large number and c is a appropriate positive number. Many methods might be used to improve the result， since the reminder term just used a relatively crude Euler-Maclaurin sum formula.
　　Dirichlet studied the divisor problem， it states to estimate the number of lattice points （x，y） with positive coordinate x，y under the hyperbola xy=R ， if we denote the number of satisfied lattice points under the hyperbola to be D（R） . Dirichlet achieved the following result：
　　D（R）=R（logR+2y-1）+O（R）， （1.2）
　　The results is also achieved by Euler-Maclaurin sum formula， and it is widely believed that the best remainder term should be R14+， where is arbitrarily small positive number.
　　In this paper， we studied the connection between the lines with irrational slope and the lattice points. We achieved the results that for a line with irrational slope， then there are infinite lattice points approximating the straight line， although the straight line may not passing through any lattice points in the plane. In this paper， we just restrict our study scope to the 2-dimension plane， every time we refer to the lines we mean a straight lines in the 2-dimension plane which built upon the Cartesian rectangular coordinate system.
　　2. Preliminaries
　　In this section， we gives some useful lemma that would be used to prove the main theorem.
　　Lemma 2.1 For a straight line y =kx+b with irrational slope k， either it pass through just one lattice points or it does not contain any lattice points.
　　Proof If the straight line is y =kx+b ， where k is the irrational number. Suppose the line contains two distinct lattice points （p1，q1）， （p2，q2） then by the slope formula we have
　　K=q2-q1p2-p1
　　where q2-q1≠ 0 and p2-p1 ≠ 0 are both integers. Then we come to conclusion that k is a rational number， which is a contradiction. So the lines with irrational slope must belongs to the kind that it pass though one lattice point ornone lattice points in the line.
　　Remark From lemma 2.1 we know that line y =kx+b with irrational slope k could not pass through more two lattice points， otherwise given any two lattice points in the line， we would come to the conclusion that the slope k is rational.
　　Lemma 2.2 For a line y =kx+b with irrational slope k and non-integral intercept b， then y =kx+b contains no lattice point.
　　Proof If y =kx+b satisfy the condition in lemma 2.2， and if the line contain a lattice point （x，y）=（p，q）， where p，q∈Z，p≠ 0， then b=q-bp would be irrational， since k is irrational and p，q are integers. But b is assumed to be rational. If ， then would also contradict the condition that in non-integral. Hence such line contains no lattice points. We have then accomplish the proof of the lemma. 　　Next， we come to the very famous theorem called Bolzano-Weierstrass Theorem， but here we just regard the theorem as a indispensable lemma.
　　Lemma 2.3 （Bolzano-Weierstrass） If S is bounded infinite point set in the number line， then there exist at least one cluster point for S.
　　Proof We use the bisection method to cut the bounded set， and then to found there exists at least cluster points for S. Here the cluster point may not belong to S.， but in the proof of next important lemma， we confine our scope to the compact set， So the cluster point must lying in the bounded set.
　　First of all， if S. is the bounded set， it must contained in the interval [a，b] that satisfy S[a，b] . Bisect this interval as [a，12（a+b）]，[12（a+b），b] and there at least one interval contains infinite points of S， we denote this interval [a1，b1]. We then divided the interval [a1，b1] in the same way as above， the we have one interval [a2，b2] that contains infinite number points of S. This process could indefinitely continued to give a set {[a1，b1]，[a2，b2]，…，}， each interval contains infinite many points of S， by the construction， we have
　　[a1，b1][a2，b2][a3，b3]……
　　Note that b1-a 1=12（b-a） ，b2-a 2=12（b1-a 1）=14（b-a） ，and general we have
　　bn-an=b-a2n，n∈N.
　　Then we prove that no two distinct points in this interval. Suppose there two points ， contained in the interval set and ≠ . Then we can find a very large n satisfy bn-an<-.This means it is impossible to have ∈[an，bn] and ∈[an，bn] for all value . We come to the conclusion that there exists at most one point belongs to all the intervals.
　　Due to properties of real number system， we denote the point to belongs to all the interval. We show that is a cluster point for S， and this proves the lemma. For sufficiently large value n， for arbitrarily small value δ we could find n satisfy bn-an<δ， then [an，bn]（-δ，+δ）. Since [an，bn] contains infinitely many points of S，then （-δ，+δ） contains at least one distinct point other than in the bounded set S. Thus is a cluster point for S， we have proved the lemma.
　　Remark There are Bolzano-Weierstrass Theorem for Rn， and the proof is more complicated than in the number line. The key point of this theorem relies in the fact that the real number system belongs to continuum， a very famous hypothesis built by German mathematician Cantor. The interested reader could consult a advanced book in real analysis.
　　We then come to the very important lemma that could be used to directly prove the main theorem in this paper. 　　Lemma 2.4 Let k be any irrational number， take c be any positive real number， and >0 be any arbitrarily small number. Then we can always find a pair of integers （p1，q1） such that
　　c　　that is，
　　0　　Similarly， we could find a pair of integers （p2，q2） satisfy
　　c-　　that is，
　　-　　Proof It is enough to prove the special case where k=a，c=y satisfy the conditions 0　　k=[k]+a where 0a<1，
　　where a is a irrational number and [k] represent the integral part of k. Similarly， we could write
　　c=[c] +y， where 0y<1，
　　If we could find integers p1，q such that
　　y　　　　then we could substitute a=k-[k]，y=c-[c]， we have
　　c-[c]　　　　Adding both side with integer [c] gives
　　c　　　　We now take q1=p1[a]+q-[c] and obtain the inequality：
　　c　　　　We prove the lemma for irrational a，y， where 0　　1. Integer multiple of length a marked on Г
　　We choose a starting point A in the circumference， and moving along the circumference， we mark out lengths a，2a，3a，…， where a represent the length of arc AB. Now， we have ja means it moves aroundthe Г for a certain number of times， for example n1=[ja] times， plus some extra distance a1， where a1 could be use the arc length in the circumference Г to represent. It has the form
　　ia=n1+a1，0　　where i，n1=[ia] are integers.
　　Similarly， we have another distinct point in the form
　　ja=n2+a2，0　　where n2[ja] is a integer. suppose ia，ja coincide in the circumference， then we arrive at
　　a=n1-n2i-j，where j≠k，（2.7）
　　thus making a a rational number， it is a contradiction to our condition. Thus we come to the conclusion that for any distinct point ia，ja ， where i≠j， represent different point in the circumference.
　　As a，2a，3a，…… moves around the circumference， there are infinite points in the bounded set Г， where every ia is different. Since Г could be parameterized by r（θ）=12πe2πiθ， where a∈[0，1]. So apply lemma 2.3， we have in the circumference Г ， there must exists at least one cluster point in Г， since Г is a compact bounded set. We call such cluster point as Q. 　　As is illustrated in Figure 2， in the neighborhood of Q， there must exists at least two distinct points β1≠β2，
　　m1a=k1+β1，
　　m2a=k2+β2，
　　measured by arc length in Г less than any given <0， where <0 is arbitrarily small positive number.
　　2. Q， Г and the points i1 on Г
　　We choose 0<1<， take β1>β2， then
　　1=β1-β2=（m1a-k1）-（m2a-k2）=ma-k（2.8）
　　We now depict a new picture about the circumference， starting from the point A， measure by the arc length in Г， where 0y<1 is arbitrarily fixed number. We also mark off the points from A at distance 1，21，3epsilon1，…. Then we would have
　　（j-1）1y　　As illustrated in Fig 2， we could find integers j′，j such that
　　y　　y-　　Since 1=ma-k， then we arrive at
　　y　　y-　　Take p1=jm，q1=jk，p2=jm，q2=j′m，q2=j′k，， then we have
　　y　　y-　　This complete the proof of the lemma.
　　3. The Main Theorem
　　In this section， we first state the core theorem in this paper， and then gives a beautiful and elaborated proof of the main theorem.
　　Theorem 3.1 For any straight line y=kx+b with k is a irrational number and b is any real number， has on either side an infinite number of lattice points lying closer to the line than any assigned distance ， where >0 could be chosen arbitrarily small.
　　Proof As is illustrated in Figure 3， suppose 1 is the fixed small positive number. If b<0 and take c=-b， applying （2.1′）， we find a lattice point p1：（p1，q1） such that 0　　0　　Then since d1>0， the point p1 lies below the line.
　　On the other hand， if b>0， we can find an n∈Z+ satisfy b-n<0， take c=-（b-n）. Then we also applying （2.1′）， we have （p2，q2） satisfy
　　0　　Thus，
　　0　　=y-（q2+n）<1（3.3）
　　And we have find a lattice point p2（p2，q2+n） lying below the line. If we denote the perpendicular distance between pi and the straight lines l to be δi， we have come to the conclusion that 0<δi<1.
　　3. Lattice points around l
　　If we picking a sequence of decreasing positive numbers 1>2>3…>0， we can construct an infinite sequence of lattice points p1，p2，p3，…， all these lattice points lying below the line and approximating the line y=kx+bat any assigned distance.
　　Similarly， by applying （2.2′ ）and use the same argument as above， we could construct the sequence of lattice points p1′，p2′，p3′，… which lying above the line l， and also approximating the line l at any assigned distance. qquad 　　4. Conclusion
　　We give some conclusion about the Main Theorem above. We have find that the line with irrational slope have such elegant properties that plenty of lattice assemble around the line， and the most amazing fact is that they are equally distributed. From the view of geometric point， there are infinite many points around the line， and it seems a very spectacular scene. Jacob Bernoulli have ever said：“It seems that a finite mind cannot comprehend the infinite". The reader could remind of the Least Square Methodfrom the statistics， it states as for given an appropriate discrete point set， we need to find the straight which makes the distance from the line to the point set have the nearest approximating distance as a whole. Conversely， if we take finite but verylarge number of points from the given lattice points in the above situation， then the straight line that satisfy the condition must be the line with irrational slope which we have given. This problem of lattice points and line with irrational slope seemsto have close relation with Least Square Method. From the viewpoint of geometry， the Main Theorem is also a remarkable theorem that should be thought in our mind.
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　　[0][99] ===enumi -0.8 em
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