Discrete Mathematics 336 (2014) 37–40

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Discrete Mathematics

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Note

A note on distinct distances in rectangular lattices

Javier Cilleruelo a, Micha Sharir b, Adam Sheffer b,∗a Instituto de Ciencias Matematicas (CSIC-UAM-UC3M-UCM), and Departamento de Matematicas, Universidad Autónoma de Madrid,28049 Madrid, Spainb School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel

a r t i c l e i n f o

Article history:Received 28 February 2014Received in revised form 25 July 2014Accepted 26 July 2014

Keywords:Discrete geometryDistinct distancesLattice

a b s t r a c t

In his famous 1946 paper, Erdős (1946) proved that the points of a√n×

√n portion of the

integer lattice determine Θ(n/√log n) distinct distances, and a variant of his technique

derives the same bound for√n ×

√n portions of several other types of lattices (e.g., see

Sheffer (2014)). In this note we consider distinct distances in rectangular lattices of theform (i, j) ∈ Z2

| 0 ≤ i ≤ n1−α, 0 ≤ j ≤ nα, for some 0 < α < 1/2, and show that

the number of distinct distances in such a lattice is Θ(n). In a sense, our proof ‘‘bypasses’’a deep conjecture in number theory, posed by Cilleruelo and Granville (2007). A positiveresolution of this conjecture would also have implied our bound.

© 2014 Elsevier B.V. All rights reserved.

Given a set P of n points in R2, let D(P )denote the number of distinct distances that are determined by pairs of pointsfrom P . Let D(n) = min|P |=n D(P ); that is, D(n) is the minimum number of distinct distances that any set of n points inR2 must always determine. In his celebrated 1946 paper [4], Erdős derived the bound D(n) = O(n/

√log n) by considering

a√n ×

√n integer lattice (a variant of his technique derives the same bound for several other types of lattices; e.g., see

Sheffer [11]). Recently, after 65 years and a series of progressively larger lower bounds,1 Guth and Katz [8] provided analmost matching lower bound D(n) = Ω(n/ log n).

While the problem of finding the asymptotic value of D(n)is almost completely solved, hardly anything is known aboutwhich point sets determine a small number of distinct distances. Consider a set P of n points in the plane, such that D(P) =

O(n/√log n). Erdős conjectured [6] that any such set ‘‘has lattice structure’’. A variant of a proof of Szemerédi implies that

there exists a line that contains Ω(√log n) points of P (Szemerédi’s proof was communicated by Erdős in [5] and can be

found in [9, Theorem 13.7]). A recent result of Pach and de Zeeuw [10] implies that any constant-degree curve that con-tains no lines and circles cannot be incident to more than O(n3/4) points of P . Another recent result by Sheffer, Zahl, and deZeeuw [12] implies that no line can contain Ω(n7/8) points of P , and no circle can contain Ω(n5/6) such points.

In this note wemake some progress towards the understanding of the structure of such sets, by showing that rectangularlattices cannot have a sublinear number of distinct distances. Specifically, we consider the number of distinct distances thatare determined by an n1−α

× nα integer lattice, for some 0 < α ≤ 1/2. We denote this number by Dα(n).The case α = 1/2 is the case of the square

√n×

√n lattice, which determinesD1/2(n) = Θ(n/

√log n) distinct distances,

as already mentioned above. Surprisingly, we show here a different estimate for α < 1/2.

Thework by Javier Cilleruelo has been supported by grantsMTM2011-22851 ofMICINN and ICMAT Severo Ochoa project SEV-2011-0087. Thework byAdam Sheffer andMicha Sharir has been supported by Grants 338/09 and 892/13 from the Israel Science Fund, by the Israeli Centers of Research Excellence(I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.∗ Corresponding author.

E-mail addresses: franciscojavier.cilleruelo@uam.es (J. Cilleruelo), michas@tau.ac.il (M. Sharir), adamsh@gmail.com, sheffera@tau.ac.il (A. Sheffer).1 For a comprehensive list of the previous bounds, see [7] and http://www.cs.umd.edu/~gasarch/erdos_dist/erdos_dist.html (version of February 2014).

http://dx.doi.org/10.1016/j.disc.2014.07.0210012-365X/© 2014 Elsevier B.V. All rights reserved.

38 J. Cilleruelo et al. / Discrete Mathematics 336 (2014) 37–40

Theorem 1. For α < 1/2, the number of distinct distances that are determined by an n1−α× nα integer lattice is

Dα(n) = n + o(n).

Proof. We consider the rectangular lattice

Rα(n) = (i, j) ∈ Z2| 0 ≤ i ≤ n1−α, 0 ≤ j ≤ nα

.

Notice that every distance between a pair of points of Rα(n) is also spanned by (0, 0) and another point of Rα(n). This im-mediately implies Dα(n) ≤ n + O(n1−α). In the rest of the proof we derive a lower bound for Dα(n). For this purpose, weconsider the sublattice

R′

α(n) = (i, j) ∈ Z2| 2nα

≤ i ≤ n1−α, 0 ≤ j ≤ nα;

since α < 1/2, R′α(n) = ∅ for n ≥ n0(α), a suitable constant depending on α. We also consider the functions

r(m) =(i, j) ∈ R′

α(n) | i2 + j2 = m,

d(m) =(i, j) ∈ R′

α(n) | i2 − j2 = m.

Observe that the smallest (resp., largest) value ofm for which d(m) = 0 is 3n2α (resp., n2−2α).We have the identities

m

r(m) =

m

d(m), (1)m

r2(m) =

m

d2(m). (2)

The identity (1) is trivial. To see (2) we observe that the sum

m r2(m) counts the number of ordered quadruples (i, j, i′, j′),for (i, j), (i′, j′) ∈ R′

α(n), such that i2 + j2 = i′2 + j′2. But this quantity also counts the number of those ordered quadruples(i, j, i′, j′), for (i, j′), (i′, j) ∈ R′

α(n), such that i2 − j′2 = i′2 − j2, which is the value of the sum

m d2(m). Putting (1) and (2)together we have

m

r(m)

2

=

m

d(m)

2

. (3)

WritingMk for the set of thosem with r(m) = k, we have

k k|Mk| = |R′α(n)|. On the other hand,

Dα(n) ≥

k≥1

|Mk|

=

k≥1

k|Mk| −

k≥1

(k − 1)|Mk|

= |R′

α(n)| −

k≥2

(k − 1)|Mk|.

Thus Dα(n) ≥ n − O(n2α+ n1−α) −

k≥2(k − 1)|Mk|. Using the inequality k − 1 ≤

k2

and (3), we have

k≥2

(k − 1)|Mk| ≤

k≥2

k2

|Mk| =

m

r(m)

2

=

m

d(m)

2

.

Theorem 1 is therefore a trivial consequence of the following proposition.

Proposition 2.m

d(m)

2

= O

n2α log2 n

.

Proof. We need the following easy lemma.

Lemma 3. If a positive integer m can be written as the product of two integers in two different ways, say m = m1m2 = m3m4,then there exists a quadruple of positive integers (s1, s2, s3, s4) satisfying

m1 = s1s2, m2 = s3s4, m3 = s1s3, m4 = s2s4.

Proof. Since m1 divides m3m4, we have m1 = s1s2 for some s1 | m3 and some s2 | m4. Putting s3 = m3/s1 and s4 = m4/s2,we havem2 = s3s4,m3 = s1s3, andm4 = s2s4.

J. Cilleruelo et al. / Discrete Mathematics 336 (2014) 37–40 39

We writem

d(m)

2

=

1≤l≤n1−2α

m∈Il

d(m)

2

,

where Il = [l2n2α, (l + 1)2n2α). We observe that the union of the intervals, namely [n2α, (1 + n1−2α)2n2α), covers all thepossiblem with d(m) = 0.

Nowwe estimate

m∈Il

d(m)

2

for a fixed l, by viewing the binomials as counting unordered pairs of distinct pairs whose

difference of squares is m. Let a2 − b2 = c2 − d2 (a > c and b > d) be such a pair of distinct representations of some m,which is counted in the above sum

m∈Il

d(m)

2

. Since m ∈ Il we have

l2n2α≤ a2 − b2 < (l + 1)2n2α.

Thus,

l2n2α≤ a2 < (l + 1)2n2α

+ b2 ≤ ((l + 1)2 + 1)n2α < (l + 2)2n2α.

The same inequality holds for c , so we have

lnα≤ a, c < (l + 2)nα. (4)

Applying Lemma 3 to (a − c)(a + c) = (b − d)(b + d) (clearly, the two products are distinct), we obtain a quadruple ofintegers (s1, s2, s3, s4) satisfying

s1s2 = a − c, s3s4 = a + c,s1s3 = b − d, s2s4 = b + d.

Using (4) and 0 ≤ b, d ≤ nα we have the following inequalities:

1 ≤ s1s2, s1s3, s2s4 ≤ 2nα,

2lnα≤ s3s4 < (2l + 4)nα. (5)

It is clear from the above inequalities that si ≤ 2nα , for i = 1, . . . , 4. From s2s4 ≤ 2nα, s1s3 ≤ 2nα , and 2lnα≤ s3s4, we

also deduce that

1 ≤ s2 ≤s3l

and 1 ≤ s1 ≤s4l. (6)

Choose s3 between 1 and 2nα . Then choose s4, according to (5), in the range2lnα

s3, (2l+4)nα

s3

. Then choose s1 and s2, according

to (6), in s3l ·

s4l ≤

(2l+4)nα

l2ways. The overall number of quadruples (s1, s2, s3, s4) under consideration is thus at most

1≤s3≤2nα

4nα

s3·(2l + 4)nα

l2= O

n2α log n

l

.

Finally we have

m

d(m)

2

≤

1≤l≤n1−2α

m∈Il

d(m)

2

= O

l≤n1−2α

n2α log nl

= On2α log2 n

.

Discussion. Theorem 1 is closely related to a special case of a fairly deep conjecture in number theory, stated as Conjecture 13in Cilleruelo and Granville [3]. This special case, given in [3, Eq. (5.1)], asserts that, for any integer N , and any fixed β < 1/2,(a, b) ∈ Z2

| a2 + b2 = N, |b| < Nβ ≤ Cβ ,

where Cβ is a constant that depends onβ (but not onN). A simple geometric argument shows that this is true forβ ≤ 1/4 butit is unknown for any 1/4 < β < 1/2. If that latter conjecture were true, a somewhat weaker version of Theorem 1 wouldfollow. Indeed, let N be an integer that can be written as i2 + j2, for 1

2n1−α

≤ i ≤ n1−α and j ≤ nα . Then N = Θ(n2(1−α)),and j = O(Nβ), for β = α/(2(1 − α)) < 1/2.

Conjecture 13 of [3] would then imply that the number of pairs (i, j) as above is at most the constant Cβ . In other words,each of the Θ(n) distances in the portion of Rα(n) with i ≥

12n

1−α , interpreted as a distance from the origin (0, 0), can beattained at most Cβ times. Hence Dα(n) = Θ(n), as asserted in Theorem 1.

The general form of Conjecture 13 [3] asserts that the number of integer lattice points on an arc of length Nβ on the circlea2 + b2 = N is bounded by some constant Cβ , for any β < 1/2. Cilleruelo and Córdoba [2] have proved this for β < 1/4.See also Bourgain and Rudnick [1] for some consequences of this conjecture.

40 J. Cilleruelo et al. / Discrete Mathematics 336 (2014) 37–40

A heuristic argument that supports the above conjecture is the following: it is well known that the quantity r(N), thatcounts the number of lattice points on the circle x2 + y2 = N , satisfies r(N) ≪ Nε for any ε > 0. If the lattice points weredistributed at random along the circle, an easy calculation would show that the probability that an arc of length Nβ containsk lattice points is bounded by

r(N)

k

N (k−1)(β−1/2). Now, for anyβ < 1/2, there exists k such that the infinite sum

N

r(N)

k

N (k−1)(β−1/2) converges, and the Borel–Cantelli lemma would then imply that, with probability 1, only a finite number ofcircles can contain k lattice points on arcs of length Nβ .

Acknowledgment

The authors would like to thank Zeev Rudnick for useful discussions on some of the number-theoretic issues.

References

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