The Compression method and applications

In this paper we introduce and develop the method of compression of points in space. We introduce the notion of the mass, the rank, the entropy, the cover and the energy of compression. We leverage this method to prove some class of inequalities related to Diophantine equations. In particular, we show that for each $L<n-1$ and for each $K>n-1$, there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align}\frac{1}{K^{n}}\ll \prod \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{\log (\frac{n}{L})}{nL^{n-1}}\nonumber \end{align}and that for each $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ and some $s\geq 2$ such that \begin{align}\sum \limits_{j=1}^{n}\frac{1}{x_j^s}\gg s\frac{n}{L^{s-1}}.\nonumber \end{align}


Introduction
The Erdós-Straus conjecture is the assertion that for each n ∈ N for n ≥ 3 there exist some x 1 , x 2 , x 3 ∈ N such that 1 More formally the conjecture states Conjecture 1.1.For each n ≥ 3, does there exist some x 1 , x 2 , x 3 ∈ N such that 1 Despite its apparent simplicity, the problem still remain unresolved.However there has been some noteworthy partial results.For instance it is shown in [2] that the number of solutions to the Erdós-Straus Conjecture is bounded polylogarithmically on average.The problem is also studied extensively in [3] and [4].The Erdós-Straus conjecture can also be rephrased as a problem of an inequality.That is to say, the conjecture can be restated as saying that for all n ≥ 3 the inequality holds for c 1 = c 2 = 4 3 for some x 1 , x 2 , x 3 ∈ N 3 .Motivated by this version of the problem, we introduce the method of compression.This method comes somewhat close to addressing this problem and its variants.Using this method, we managed to show that Theorem 1.1.For each L ∈ N with L > n − 1 there exist some (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n such that for some c 1 , c 2 > 1.In particular, for each L ≥ 3 there exist some (x 1 , x 2 , x 3 ) ∈ N 3 with x 1 = x 2 , x 2 = x 3 and x 3 = x 1 such that Perhaps more general is the result Theorem 1.2.For each L > n − 1 there exist some (x 1 , x 2 , . . ., x n ) with x i = x j for all 1 ≤ i < j ≤ n and some s ≥ 2 such that Theorem 1.3.For each L < n−1 and for all s ≥ 2, there exist some (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for 1 ≤ i < j ≤ n such that n j=1 1 x s j ≪ log s n L .

Compression
Definition 2.1.By the compression of scale 1 ≥ m > 0 on R n , we mean the map for n ≥ 2 and with x i = 0 for all i = 1, . . ., n.
Remark 2.2.The notion of compression is in some way the process of rescaling points in R n for n ≥ 2. Thus it is important to notice that a compression pushes points very close to the origin away from the origin by certain scale and similarly draws points away from the origin close to the origin.Intuitively, one could think of a compression as inducing a certain kind of motion on points in the Euclidean space f any dimension.
Proposition 2.1.A compression of scale 1 ≥ m > 0 with V m : R n −→ R n is a bijective map.
Proof.Suppose V m [(x 1 , x 2 , . . ., x n )] = V m [(y It follows that x i = y i for each i = 1, 2, . . ., n. Surjectivity follows by definition of the map.Thus the map is bijective.Proposition 3.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n with x i = x j for each i = j, then the estimates holds for n ≥ 2.
Proof.Let (x 1 , x 2 , . . ., x n ) ∈ R n for n ≥ 2 with x j ≥ 1.Then it follows that and the upper estimate follows by the estimate for this sum.The lower estimate also follows by noting the lower bound The estimates obtained for the mass of compression is quite suggestive.It restricts the entries of any of our choice of tuple to be distinct.After a little heuristics, It can be seen the left estimate for the mass of compression tends to be almost flawed if we allow for tuples with at least two similar entries.Thus in building this Theory, and with all the results we will obtained, we will enforce that the entries of any choice of tuple is distinct.

Application of mass of compression.
In this section we apply the notion of the mass of compression to the Erdós-Straus conjecture.
The lower bound also follows by noting that and the inequality follows by taking sup(x j ) = L 1 and Inf(x j ) = L 2 .Theorem 3.3 is redolent of the Edòs-Strauss conjecture.Indeed It can be considered as a weaker version of the conjecture.It is quite implicit from Theorem 3.3 that there are infinitely many points in N n that satisfy the inequality with finitely many such exceptions.Therefore in the opposite direction we can assert that there are infinitely many L 1 , L 2 ∈ N that satisfies the inequality.We state a consequence of the result in Theorem 3.3 to shed light on this assertion.Corollary 3.1.For each L ∈ N with L > n − 1 there exist some (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n such that In particular, for each L ≥ 3 there exist some ( By taking K = sup(x j ) and L = Inf(x j ) for any such points, it follows that The special case follows by taking n = 3.
It is important to recognize that the condition (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j all 1 ≤ i < j ≤ n in the statement of the result is not only a quantifier but a requirement; otherwise, the estimate for the mass of compression will be flawed completely.To wit, suppose that we take x 1 = x 2 = . . .= x n , then it will follow that Inf(x j ) = sup(x j ), in which case the mass of compression of scale m satisfies and it is easy to notice that this inequality is absurd.By extension one could also try to equalize the sub-sequence on the bases of assigning the supremum and the infimum and obtain an estimate but that would also contradict the mass of compression inequality after a slight reassignment of the sub-sequence.Thus it is important for the estimates to make any good sense to ensure that any tuple (x 1 , x 2 , . . ., x n ) ∈ N n must satisfy x i = x j for all 1 ≤ i < j ≤ n.Thus our Theory will be built on this assumption, that any tuple we use has to have distinct entry.Since all other statistic will eventually depend on the mass of compression, this assumption will be highly upheld.
Remark 3.4.The result can be interpreted as saying that for each L ≥ 3 there exist some (x 1 , x 2 , x 3 ) ∈ N 3 such that Investigating the scale of these constants is the motivation for this Theory and will be developed in the following sequel.Theorem 3.5.For each K > n − 1 and for each L < n − 1, there exist some (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n such that Proof.Let us choose (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n such that Inf(x j ) < n − 1 and sup(x j ) > n − 1.Then we set L = Inf(x j ) and K = sup(x j ), then the result follows from the estimate in Theorem 3.1.
Remark 3.6.Next we expose one consequence of Theorem 3.5.

The rank of compression
In this section we introduce the notion of the rank of compression.We launch the following language in that regard.Definition 4.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n for n ≥ 2 then by the rank of compression, denoted R, we mean the expression Remark 4.2.It is important to notice that the rank of a compression of scale 1 ≥ m > 0 is basically the distance of the image of points under compression from the origin.Next we relate the rank of compression of scale 1 ≥ m > 0 with the mass of a certain compression of scale 1.
Remark 4.3.Next we prove upper and lower bounding the rank of compression of scale 1 ≥ m > 0 in the following result.We leverage pretty much the estimates for the mass of compression of scale 1 ≥ m > 0.

Application of rank of compression.
In this section we expose one consequence of the rank of compression.We apply this to estimate the second moment unit sum of the Erdós Type problem.We state this more formally in the following result.
Theorem 4.5.For each L > √ n − 1, there exist some (x 1 , x 2 , . . ., x n ) ∈ N n with In particular for each L ≥ 2, there exist some (x 1 , x 2 , x 3 ) ∈ N 3 with x 1 = x 2 , x 2 = x 3 and x 1 = x 3 and some constant c 1 , c 2 > 1 such that Then the inequality follows immediately.The special case follows by taking n = 3.
Remark 4.6.Next we present a second moment variant inequality of the unit sum of positive integers in the following statement.

The entropy of compression
In this section we launch the notion of the entropy of compression.Intuitively, one could think of this concept as a criteria assigning a weight to the image of points under compression.We provide some quite modest estimates of this statistic and exploit some applications, in the context of some Diophantine problems.Definition 5.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n with x i = 0, 1 for all i = 1, 2 . . ., n.By the entropy of a compression of scale 1 ≥ m > 0, we mean the map E : R n −→ R such that Remark 5.2.Next we relate the mass of a compression to the entropy of compression and deduce reasonable good bounds for our further studies.We could in fact be economical with the bounds but they are okay for our needs.
Proposition 5.1.For all n ≥ 2, we have Proof.By Definition 3.1, we have The result follows immediately from this relation.
Proof.The result follows by using the relation in Proposition 5.1 and leveraging the bounds in Proposition 3.1, and noting that

Applications of the entropy of compression.
In this section we lay down one striking and a stunning consequence of the entropy of compression.One could think of these applications as analogues of the Erdós type result for the unit sums of triples of the form (x 1 , x 2 , x 3 ).We state two consequences of these estimates in the following sequel.
Theorem 5.3.For each L > n − 1, there exist some Theorem 5.3 tells us that for some tuple (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n there must exist some constant c 1 , c 2 > 1 such that we have the inequality Next we present a second application of the estimates of the entropy of compression in the following sequel.
Theorem 5.4.For each L < n − 1 and for each K > n − 1, there exist some Proof.Let us choose a tuple (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n such that sup(x j ) = K > n − 1 and L = Inf(x j ) < n − 1, then the result follows immediately.
Corollary 5.1.For each L < 4 and for each K > 4, there exist some (x 1 , x 2 , x 3 , x 4 , x 5 ) ∈ N 5 with x i = x j for all 1 ≤ i < j ≤ 5 and some constant c 1 , c 2 > 1 such that Proof.The result follows by taking n = 5 in Theorem 5.3.

Compression gap
In this section we introduce the notion of the gap of compression.We investigate this concept in-depth and in relation to the already introduced concepts.
The gap of compression is a definitive measure of the chasm between points and their image points under compression.We can estimate this chasm by relating the compression gap to the mass of an expansion in the following ways.Proposition 6.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n for n ≥ 2 with x j = 0, 1 for j = 1, . . ., n, then we have Proof.The result follows by using using Definition 6.1 and Definition 3.1.
Remark 6.2.We are now ready to provide an estimate for the gap of compression.
Proof.The result follows by exploiting Proposition 3.1 in Proposition 9.1 and noting that

The energy of compression
In this section we introduce the notion of the energy of compression.We launch more formally the following language.Definition 7.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n with x i = 0, 1 for all i = 1, 2 . . ., n for n ≥ 2, then by the energy dissipated under compression on (x 1 , x 2 , . . ., x n ), denoted E, we mean the expression Remark 7.2.Given that we have obtained upper and lower bounds for the compression gap and the entropy of any points under compression, we can certainly get control on the energy dissipated under compression in the following proposition. and Proof.The result follows by plugging the estimate in 6.3 and 5.2 into definition 7.1.
7.1.Applications of the energy of compression.In this section we give some consequences of the notion of the energy of compression.
Theorem 7.3.For each K > n − 1 and for each L < n − 1, there exist some Proof.First choose (x 1 , x 2 , . . ., x n ) ∈ N n with x i = x j for all 1 ≤ i < j ≤ n such that Inf(x j ) < n − 1 and sup(x j ) > n − 1.Now set K = sup(x j ) and Inf(x j ) = L, then the result follows by exploiting the estimates in Proposition 7.1.
Corollary 7.1.For each K ≥ 5 and for each L < 4, there exist some (x 1 , x 2 , x 3 , x 4 , x 5 ) ∈ N 5 with x i = x j for all 1 ≤ i < j ≤ 5 such that Proof.The result follows by taking n = 5 in Theorem 7.3.

The measure and cost of compression
In this section we introduce the notion of the measure and the cost of compression.We launch the following languages.Definition 8.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n with x i = 0, 1 for all i = 1, 2 . . ., n for n ≥ 2. Then by the measure of compression on (x 1 , x 2 , . . ., x n ), denoted N , we mean the expression The corresponding cost of compression, denoted by C, is given Next we estimate from below and above the measure and the cost of compression in the following sequel.We leverage the estimates established thus far to provide these estimates.Proposition 8.1.Let (x 1 , x 2 , . . ., x n ) ∈ N n , then we have the following bounds Proof.The result follows by exploiting the estimates in Theorem 5.2 in definition 8.1.
Proof.The result follows by leveraging various estimates developed.

The ball induced by compression
In this section we introduce the notion of the ball induced by a point (x 1 , x 2 , . . ., x n ) ∈ R n under compression of a given scale.We launch more formally the following language.
Definition 9.1.Let (x 1 , x 2 , . . ., x n ) ∈ R n with x i = x j for all 1 ≤ i < j ≤ n and x i = 0 for all 1 ≤ i ≤ n.Then by the ball induced by ( Remark 9.2.Next we prove that smaller balls induced by points should essentially be covered by the bigger balls in which they are embedded.We state and prove this statement in the following result.
In the geometry of balls induced under compression of scale m > 0, we assume implicitly that 0 < m ≤ 1.
For simplicity we will on occasion choose to write the ball induced by the point x = (x 1 , x 2 , . . ., x n ) under compression as . We adopt this notation to save enough work space in many circumstances.We first prove a preparatory result in the following sequel.We find the following estimates for the compression gap useful.
In particular, if m = m(n) = o(1) as n −→ ∞, then we have the estimate Proposition 9.1 offers us an extremely useful identity.It allows us to pass from the gap of compression on points to the relative distance to the origin.It tells us that points under compression with a large gap must be far away from the origin than points with a relatively smaller gap under compression.That is to say, the inequality || y|| for x, y ∈ R n with x i ≥ 1 for all 1 ≤ i ≤ n.This important transference principle will be mostly put to use in obtaining our results.In particular, we note that in the latter case, we can write the asymptotic with || y − z|| < ǫ for some ǫ > 0 then it follows that || y|| || z||, which is absurd.In this case, we can take ǫ := then it follows from Proposition 9.1 that || z|| || y||.Under the requirement || y − z|| < ǫ for some ǫ > 0, we obtain the inequality [ y] and the proof of the theorem is complete.
In the geometry of balls under compression, we will assume that n is sufficiently large for R n .In this regime, we will always take the scale of compression m := m [ y] by virtue of their compression gaps and the latter does not contain the point x by construction.It is easy to see that [ x] and this completes the proof.Remark 9.6.Theorem 9.5 tells us that points confined in certain balls induced under compression should by necessity have their induced ball under compression covered by these balls in which they are contained.9.1.Interior points and the limit points of balls induced under compression.In this section we launch the notion of an interior and the limit point of balls induced under compression.We study this notion in depth and explore some connections.Definition 9.7.Let y = (y 1 , y 2 , . . ., y n ) ∈ R n with y i = y j for all 1 ≤ i < j ≤ n.
An interior point z then said to be a limit point if [ y] Remark 9.8.Next we prove that there must exist an interior and limit point in any ball induced by points under compression of any scale in any dimension.Theorem 9.9.Let x = (x 1 , x 2 , . . ., x n ) ∈ R n with x i = x j for all 1 ≤ i < j ≤ n with y i ≥ 1 for all 1 ≤ i ≤ n.Then the ball B 1 2 G•Vm[ x] [ x] contains an interior point and a limit point.
Proof.Let x = (x 1 , x 2 , . . ., x n ) ∈ R n with x i = x j for all 1 ≤ i < j ≤ n with x i ≥ 1 for all 1 ≤ i ≤ n and suppose on the contrary that B 1 2 G•Vm[ x] [ x] contains no limit point.Then pick for ǫ > 0 sufficiently small.Then by Theorem 9.5 and Theorem 9.4, it follows that Then employing Theorem 9.5 and Theorem 9.4, we have . By continuing the argument in this manner we obtain the infinite descending sequence of the gap of compression [ x] contains only the point x.It follows by Definition 9.9 the point x must be the limit point of the ball [ y] for any y = (y 1 , y 2 , . . ., y n ) ∈ R n with y i > 1 for all 1 ≤ i ≤ n.For if the contrary [ y] holds for some y = (y 1 , y 2 , . . ., y n ) ∈ R n with y i > 1 for each 1 ≤ i ≤ n, then there must exists some point [ y] which is inconsistent with the fact that x is the limit point of the ball.9.2.Admissible points of balls induced under compression.We launch the notion of admissible points of balls induced by points under compression.We study this notion in depth and explore some possible connections.Definition 9.10.Let y = (y 1 , y 2 , . . ., y n ) ∈ R n with y i = y j for all 1 ≤ i < j ≤ n.Then y is said to be an admissible point of the ball Remark 9.11.It is important to notice that the notion of admissible points of balls induced by points under compression encompasses points on the ball.These points in geometrical terms basically sit on the outer of the induced ball.Next we show that all balls can in principle be generated by their admissible points.

This already contradicts the equality
. The latter equality of compression gaps follows from the requirement that the balls are indistinguishable.Conversely, suppose Then it follows that the point y lives on the outer of the two indistinguishable balls and so must satisfy the equality and y is indeed admissible, thereby ending the proof.

Application to the Erdős unit distance problem
Erdős posed in 1946 the problem of counting the number of unit distances that can be determined by a set of n points in the plane.It is known (see [6]) that the number of unit distances that can be determined by n points in the plane is lower bounded by Erdős asks if the upper bound for the number of unit distances that can be determined by n points in the plane can also be a function of this form.In other words, the problem asks if the lower bound of Erdős is the best possible.What is known currently is the upper bound (see [7]) proportional to the quantity as the translation of the ball by the vector v ∈ R k , so that for any 2 and apply the compression V m on x j .Next construct the ball induced under compression We remark that the ball so constructed is a ball of radius 1  2 G • V m [ x j ] = 1 2 , so that for any admissible point so that any such n 2 of admissible points determines at least n 2 unit distances.Now for any n such admissible points on the ball and by virtue of the restriction we make the optimal assignment max 1≤j≤n sup 1≤s≤2 (x js ) = n o(1) , since points x l far away from the origin with x ls for 1 ≤ s ≤ 2 must have large compression gaps by virtue of Lemma 9.3 and the ensuing discussion.In particular, the point x l must be such that x ls = 1 + ǫ with 1 ≤ s ≤ 2 for any small ǫ > 0 in order to satisfy the requirement in (10.1).The number of unit distances induced by n admissible points on the ball so constructed is at most (1) .Now for any set of n points in general position in the plane R 2 , let us apply the translation with a fixed vector v ∈ R 2 now lives in the smallest region containing all the n points in general position.We remark that this new ball is still of radius 1 2 but contains points -including admissible points -all of which are translates of points in the previous ball ] by a fixed vector v ∈ R 2 .We remark that the unit distances are all preserved so that the number of unit distances determined by the n points in general position is upper bounded by ≪ 2 n 1+o (1)   thereby ending the proof.

Application to counting integral points in a circle and a grid
The Gauss circle problem is a problem that seeks to counts the number of integral points in a circle centered at the origin and of radius r.It is fairly easy to see that the area of a circle of radius r > 0 gives a fairly good approximation for the number of such integral points in the circle, since on average each unit square in the circle contains at least an integral point.In particular, by denoting N (r) to be the number of integral points in a circle of radius r, then the following elementary estimate is well-known where |E(r)| is the error term.The real and the main problem in this area is to obtain a reasonably good estimate for the error term.In fact, it is conjectured that for ǫ > 0. The first fundamental progress was made by Gauss [9], where it is shown that G.H Hardy and Edmund Landau almost independently obtained a lower bound [1] by showing that ).
The current best upper bound (see [8]) is given by In this paper we study a variant of this problem in the region between a general k dimensional grid 2r × 2r • • • × 2r (k times) and the largest sphere contained in the grid.In particular, we obtain the following lower bound for the number of integral points in this region Remark 11.1.We now apply the method to obtain a lower bound for the number of lattice points in k-dimensional sphere of radius r > 0.
Theorem 11.2.Let N k (r) denotes the number of integral points in the k dimensional sphere of radius r > 0. Then N k (r) satisfies the lower bound 1) .
Proof.Pick arbitrarily a point (x 1 , x 2 , . . ., x k ) = x ∈ R k with x i > 1 for 1 ≤ i ≤ k and x i = x j for i = j such that G • V m [ x] = 2r.This ensures the ball induced under compression is of radius r.Next we apply the compression of fixed scale 0 < m ≤ 1 and set m = m(k) = o(1) as k −→ ∞, given by V m [ x] and construct the ball induced by the compression given by with radius (G•Vm[ x]) 2 = r.By appealing to Theorem 9.12 admissible points x l ∈ R k ( x l = x) of the ball of compression induced with || x l − x|| < ǫ for ǫ > 0 sufficiently small must satisfy the condition G • V m [ x l ] = 2r.Also by appealing to Theorem 9.4 points [ x] that are not admissible must satisfy the inequality covers this ball, we make the assignment as r −→ ∞.This ensures that points in the k-dimensional box are confined in the ball.The number of integral points in the largest ball contained in the 2r and the lower bound follows by our choice 11.1.Application to counting the number of integral points on the boundary of a k-dimensional sphere.Theorem 11.3.Let N r,k denotes the number of integral points on the boundary of a k-dimensional sphere of radius r.Then N r,k satisfies the lower bound with radius (G•Vm[ x]) 2 = r.We remark that this ball is exactly covered by the k-dimensional box 2r × 2r × • • • × 2r (k times).By appealing to Theorem 9.12 admissible points x l ∈ R k ( x l = x) of the ball of compression induced must satisfy the condition G • V m [ x l ] = 2r.The number of integral points on the boundary of the k-dimensional sphere is lower bounded by and the lower bound follows.
12. Application to the general distance problem in R k Proof.Pick arbitrarily a point (x 1 , x 2 , . . ., such that V m [ x l ] = x h so that the number of d-unit distances is lower bounded by and the lower bound follows.Proof.Pick arbitrarily a point ( . By appealing to Theorem 9.12 admissible points x l ∈ R k ( x l = x) of the ball of compression induced must satisfy the condition G • V m [ x l ] = d r .Next we count the number of d r -unit distances formed by a set of n points in R k by counting pairs of admissible points ( x l , x h ) on the ball and the lower bound follows.

Application to the Ehrhart volume conjecture
The Ehrhart volume conjecture is the assertion that any convex body K in R n with a single lattice point in it's interior as barycenter must have volume satisfying the upper bound V ol(K) ≤ (n + 1) n n! .
The conjecture has only been proven for various special cases in very specific settings.For instance, Ehrhart proved the conjecture in the two dimensional case and for simplices [11].The conjecture has also been settled for a large class of rational polytopes [10].In this paper, we study the Ehrhart volume conjecture.We show that the claimed inequality fails for some convex bodies, providing a counter example to the Ehrhart volume conjecture.The main idea that goes into the disprove pertains to a certain construction of a ball in R n and the realization that after some little tweak of the internal structure, the ball satisfies the requirements of the conjecture but has too much volume, at least a volume beyond that postulated by Ehrhart.In particular, we prove the following lower bound Theorem 14.1.Let V ol(K) denotes the volume of a ball in R n with only one lattice points in it's interior as its center of mass.Then V ol(K) satisfies the lower bound This ensures the ball induced under compression is of radius n 2 .Next we apply the compression of fixed scale 0 < m ≤ 1, given by V m [ x] with m := m(n) = o(1) as n −→ ∞ and construct the ball induced by the compression given by By appealing to Theorem 9.12 admissible points x l ∈ R k ( x l = x) of the ball of compression induced must satisfy the condition G • V m [ x l ] = n with || x l − x|| < δ for δ > 0 sufficiently small.Also by appealing to Theorem 9.4 points The number of integral points in the largest ball contained in the n × n × • • • × n (n times) grid that shares admissible points on both sides with the grid is √ n inf(x li ) n Let d > 0, then the following question appears in [12] Question 15.1.What is the maximum number of points included in a plane figure (generally: in a space body) such that the distance between any two points is greater than or equal to d?
Though it belongs to the class of discrete geometry problems involving certain configurations of points and lines in the plane (resp.Euclidean space), the problem 15.1 is relatively unknown and unsolved.Depending on the dimension of the space in which the points dwell, the problem demands a precise arrangement of points so that their mutual distances are not small and are totally covered by a planar figure (resp.space body).In theory, the problem might be investigated by selecting a planar (resp.space curve) that contains all of these points in the correct configuration, as this curve can be embedded in a planar shape (resp.space body) or its slightly expanded and translated equivalents.This is the main concept we will use to get the major result in this paper.By using the method of compression, we show that the maximum number of points that can be included in a planar figure with mutual distances at least d > 0 is at least d ǫ .In particular, we obtain the following lower bound Theorem 15.2.Let ∆ 2 (d) denotes the maximum number of points that can be placed inside a geometric figure in R 2 such that their mutual distances is at least d > 0. Then the lower bound holds for some small ǫ > 0.
Next we obtain an equivalent notion of the circumference of the circle induced by points under compression in the plane R 2 in the following result.
Proposition 15.1.Let x ∈ R 2 with x i = 0 for each 1 ≤ i ≤ 2. Then the circumference of the circle induced by point x under compression of scale 0 < m ≤ 1.Then the compression denoted V m [ x] is given by Proof.This follows from the mere definition of the circumference of a circle and noting that the radius r of the circle induced by the point x ∈ R 2 under compression is given by for sufficiently small ǫ > 0 and for some function f : R −→ R such that the circle of compression constructed lives in the plane figure.On this circle locate admissible points so that the chord joining each pair of adjacent admissible points is of length d > 0. Invoking Proposition 15.1, the circumference of the circle induced under compression is given by

3 . 3 . 1 .Remark 3 . 2 .
The mass of compression Definition By the mass of a compression of scale 0 < m ≤ 1, we mean the map M : R n −→ R such that M(V m [(x 1 , x 2 , . . ., x n )]) = Next we prove upper and lower bounding the mass of the compression of scale 0 < m ≤ 1.

n 4 3
due to Spencer, Szemeredi and Trotter.Definition 10.1 (Translation of balls).Let x ∈ R k and B 1 2 G•Vm[ x] [ x] be the ball induced under compression.Then we denote the map 2r.This ensures the ball induced under compression is of radius r.Next we apply the compression of fixed scale 0 < m ≤ 1, given by V m [ x] with m := m(k) = o(1) as k −→ ∞ and construct the ball induced by the compression given by

Theorem 12 . 1 .
Let D n,d denotes the number of d-unit distances (d > 0) that can be formed from a set of n points in R k .Then the lower bound holds D n,d ≫ n √ k d .

2 = d 2 .
for a fixed d > 0. This ensures the ball induced under compression is of radius d 2 .Next we apply the compression of fixed scale 0 < m ≤ 1, given by V m [ x] with m := m(k) = o(1) and construct the ball induced by the compression given byB 1 2 G•Vm[ x] [ x]with radius (G•Vm[ x]) By appealing to Theorem 9.12 admissible pointsx l ∈ R k ( x l = x)of the ball of compression induced must satisfy the condition G • V m [ x l ] = d.Next we count the number of d-unit distances formed by a set of n points in R k by counting pairs of admissible points ( x l , x h ) on the ball B 1 2 G•Vm[ x] [ x]

13.
Application to counting the average number of integer powered distances in R k Theorem 13.1.Let D n,d r denotes the number of d r -unit distances (d > 0) that can be formed from a set of n points in R k for a fixed r > 1.Then the lower bound holds 1≤d≤t D n,d r ≫ n 2r √ k log t for a fixed t > 1.
for a fixed d > 0 and r > 1.This ensures the ball induced under compression is of radius d r 2 .Next we apply the compression of fixed scale 0 < m ≤ 1, given by V m [ x] with m := m(k) = o(1) as k −→ ∞ and construct the ball induced by the compression given by

x
l ∈n n ⊂R n 1≤i≤n min x l ∈n n inf(x li )≫ min x l ∈n n inf(x li )We note that the number of lattice pointsN n (n) in the ball K := B 1 2 G•Vm[ x] [ x] and the volume V ol(K) satisfies the asymptotic relation N n (n) ∼ V ol(K) so that by removing all sub-grid of the grid n × n • • • × n (n times) contained in the ball K := B 1 2 G•Vm[ x] [ x] except the sub-grid n 2 × n 2 × • • • n 2 (n times), we see that we are left with only one lattice point as the center of the ball.This completes the construction.15.Application to counting the maximum number of points in a plane figure with large pairwise distances

16. Lower bound Theorem 16. 1 . 1 2 2 ≥
Let ∆ 2 (d) denotes the maximum number of points that can be placed inside a geometric figure in R 2 such that their mutual distances is at least d > 0 .Then the lower bound holds∆ 2 (d) ≫ 2 d ǫ for some small ǫ > 0. Proof.Pick arbitrarily a point (x 1 , x 2 ) = x ∈ R 2 such that G • V m [ x] ≥ d f (d) .Next we apply the compression of scale 1 ≥ m > 0, given by V m [ x] with m := m(2) = and construct the circle induced by the compression given byB 1 2 G•Vm[ x] [ x] with radius (G•Vm[ x]) d f (d) 2 by choosing sup(x i ) 1≤i≤2 = inf(x i ) 1≤i≤2 = d f (d)+ǫ 1 , y 2 , . . ., y n )], then it follows that