ON TWO-POINT CONFIGURATIONS IN RANDOM SET
HOI H. NGUYEN
Abstract. We show that with high probability a random set of size Θ(n1−1/k )
of {1, . . . , n} contains two elements a and a + dk , where d is a positive integer.
As a consequence, we prove an analogue of Sárközy-Fürstenberg’s theorem for
random set.
1. Introduction
Let ℘ be a general additive configuration, ℘ = (a, a+P1 (d), . . . , a+Pk−1 (d)), where
Pi ∈ Z[d] and Pi (0) = 0. Let [n] denote the set of positive integers up to n. A
natural question is:
Question 1.1. How is ℘ distributed in [n]?
Roth’s theorem [6] says that for given δ > 0 and sufficiently large n, any subset of
size δn of [n] contains a nontrivial sample of ℘ = (a, a + d, a + 2d) (here nontrivial
sample means d 6= 0). (In fact Roth proved for δ = O(1/ log log n)). In 1975,
Szemerédi [8] extended Roth’s theorem for general linear configurations ℘ = (a, a +
d, . . . , a + (k − 1)d).
For configuration of type ℘ = (a, a + P (d)), Sárközy [7] and Fürstenberg [2] independently discovered a similar phenomenon.
Theorem 1.2 (Sárközy and Fürstenberg’s theorem, quantitative version). [9, Theorem 3.2][4, Theorem 3.1] Let δ > 0 be a fixed positive real number, and let P
be a polynomial of integer coefficients and P (0) = 0. Then there exist an integer n = n(δ, P ) and a posisitve constant c(δ, P ) with the following property. If
n ≥ n(δ, P ), and A ⊂ [n] is any set of cardinality at least δn, then
• A contains a nontrivial sample of ℘.
• A contains at least c(δ, P )|A|2 n1/ deg(P )−1 samples of ℘.
In 1996, Bergelson and Leibman [1] extended this result for all configurations ℘ =
(a, a + P1 (d), . . . , Pk−1 (d)).
This work was written while the author was supported by a DIMACS summer research fellowship, 2008.
1
2
HOI H. NGUYEN
Following Question 1.1, one may consider distribution of ℘ in pseudo-random sets,
for instance:
Question 1.3. Does the set of primes P contain nontrivial samples of ℘? How is
℘ distributed in P?
The famous Green-Tao theorem [3] says that any subset of upper positive density
of P contains many samples of ℘ = (a, a + d, . . . , a + (k − 1)d) for any k. This
phenomenon also holds for more general configuration (a, a+P1 (d), . . . , a+Pk−1 (d))
by the work of Tao and Ziegler [9].
The main goal of this short note is working with a similar question for random sets.
Question 1.4. How is ℘ distributed in a typical random set?
For short, let us say that a set A is (δ, ℘) if any set B ⊂ A of size at least δ|A|
contains a nontrivial sample of ℘.
In 1991, Kohayakawa-ÃLuczak-Rödl [5] showed the following result.
Theorem 1.5. Almost every set R of cardinality |R| = r Àδ n1/2 of [n] is (δ, (a, a+
d, a + 2d)).
The assumption r Àδ n1/2 is tight, up to a constant factor. Indeed, a typical
random set of size r contains about Θ(r3 /n) 3-term arithmetic progressions. Thus
if (1 − δ)r À r3 /n then there is a subset of size δr which does not contain any
nontrivial arithmetic progression.
Motivated by Theorem 1.5, L
à aba and Hamel [4] studied the distribution of ℘ =
(a, a + dk ) in random sets, as follows.
Theorem 1.6. Let k ≥ 2 be a fixed integer. Then there exists a positive real
number ε(k) with the following property. Let δ > 0 be fixed, then almost every set
R of cardinality |R| = r Àδ n1−ε(k) is (δ, (a, a + dk )).
It has been shown that ε(2) = 1/110, and ε(3) À ε(2), etc. Although the method
used in [4] is strong, it seems to fall short for obtaining relatively good estimates
upon ε(k).
Heuristically, one may guess that ε(k) = 1/k. This bound, if true, would be optimal.
Indeed, a typical random set of size r contains Θ(n1+1/k r2 /n2 ) samples (a, a + dk ).
Thus if (1 − δ)r À n1+1/k r2 /n2 , i.e. r ¿δ n1−1/k , then there is a subset of size δr
which does not contain any non-trivial sample (a, a + dk ).
In this short note we shall claim this heuristic prediction, sharpening Theorem 1.6.
Theorem 1.7 (Main theorem). Almost every set of size r Àδ n1−1/k of [n] is
(δ, (a, a + dk )).
ON TWO-POINT CONFIGURATIONS IN RANDOM SET
3
Our approach is elementary: by using a combinatorial lemma and Sárközy-Fürstenberg’s
theorem.
2. A combinatorial lemma
Let G(X, Y ) be a bipartite graph. We denote e(X, Y ) the number of edges going
¯
through X and Y . The average degree of G, d(G),
is e(X, Y )/(|X||Y |).
Roughly speaking, if we choose vertices of a relatively dense “pseudo-random” graph
randomly, then the formed graph is unlikely to be empty. (This intuition was also
used in [5]).
Lemma 2.1. Let {G = G([n], [n])}∞
n=1 be a sequence of bipartite graphs. Assume
that for every ε > 0 there exist an integer n(ε) and a number c(ε) > 0 such that
¯
e(A, A) ≥ c(ε)|A|2 d(G)/n
for n ≥ n(ε) and any A ⊂ [n] such that |A| ≥ εn. Then
for any given number α > 0 there exist an integer n(α) and a number C(α) > 0 with
the following property. If one chooses a random set S of size s from [n], then the
¯
probability that G(S, S) being empty is at most αs , providing that s ≥ C(α)n/d(G)
and n ≥ n(α).
Proof (of Lemma 2.1) For brevity we write V for the ground set [n]. We shall view
S as an ordered random set, whose elements will be chosen in order, v1 first, and
vs last. We shall verify the lemma within this probabilistic model. Deduction of
the original model follows easily.
Let Nk be the set of neighbors of the first k chosen vertices, that is Nk = {v ∈
V, (vi , v) ∈ E(G) for some i ≤ k}.
Assume that G(S, S) does not contain any edge. Thus vk+1 ∈
/ Nk for every k. Let
¯
Bk+1 be the set of those vk+1 ’s of V \{v1 , . . . , vk } such that Nk+1 \Nk ≤ c(ε)εd(G),
2
where ε will be chosen to be small enough ( ε = α /6 is fine) and c(ε) is the constant
from Lemma 2.1. We observe the following.
Claim 2.2. |Bk+1 | ≤ ε|V |.
Proof (of Claim 2.2) Assume for contradiction that |Bk+1 | ≥ ε|V | = εn. Since
Bk+1 ∩ Nk = ∅, we have
2¯
¯
e(Bk+1 , Bk+1 ) ≤ e(Bk+1 , V \Nk ) ≤ c(ε)εd(G)|B
k+1 | < c(ε)|Bk+1 | d(G)/n.
which contradicts with the property of G.
Thus we infer that if G(S, S) does not contain any edge then |Bk+1 | ≤ ε|V | for
every k.
Now we assume that s is sufficiently large,
4
HOI H. NGUYEN
¯
s ≥ 2(c(ε)ε)−1 n/d(G).
Let s0 be the number of indices k + 1 such that vk+1 ∈
/ Bk+1 . Then
n ≥ |Ns | ≥
X
¯
|Nk+1 \Nk | ≥ s0 c(ε)εd(G).
vk+1 ∈B
/ k+1
Thus
¯
s0 ≤ (c(ε)ε)−1 n/d(G)
≤ s/2.
As a result, there are s − s0 vertices vk+1 which lie in Bk+1 . But |Bk+1 | ≤ εn, we
infer that the number of S such that G(S, S) is empty is bounded by
X µs¶ 0
0
ns (εn)s−s ≤ (6ε)s/2 n(n − 1) . . . (n − s + 1)
0
s
0
s ≤s/2
≤ αs n(n − 1) . . . (n − s + 1),
completing the proof.
3. Proof of Theorem 1.7
We define a bipartite graph G on [n] × [n] = V1 × V2 by connecting u ∈ V1 to
¯
v ∈ V2 if v − u = dk for some integer d ∈ [1, n1/k ]. Notice that d(G)
≈ Cn1/k for
some absolute constant C.
Let us now restate Sárközy-Fürstenberg’s theorem (Theorem 1.2, for P (d) = dk ) in
terms of graph, which very roughly says that G is pseudo-random.
Theorem 3.1. Let ε > 0 be a positive constant. Then there exist a positive integer
n(ε, k) and a positive constant c(ε, k) such that e(A, A) ≥ c(ε, k)|A|2 n1/k−1 for
n ≥ n(ε, k) and any A ⊂ [n] such that |A| ≥ εn.
Let S be a set of size s of [n]. We call S bad if it does not contain any sample of
(a, a + dk ), in other words, S is bad if G(S, S) does not contain any edge.
By Lemma 2.1 the number of bad sets is at most αs
¯
C(α)n/d(G).
This is satisfied if s ≥ 2C(α)C −1 n1−1/k .
¡n¢
s , providing that s ≥
ON TWO-POINT CONFIGURATIONS IN RANDOM SET
5
Now we let r = s/δ and consider a random set R of size r. The probability that R
contains a bad set of size s is at most
µ ¶µ
¶ µ ¶
n n−s
n
/
= o(1)
α
s
r−s
r
s
providing small enough α = α(δ).
To finish the proof, we note that if R does not contain any bad set of size δr, then
R is (δ, (a, a + dk )).
References
[1] V. Bergelson and A. Leibman, Polynomial extensions of Van Der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc. 9(1996), no. 3, 725-753.
[2] H. Fürstenberg Recurrence in ergodic theory and combinatorial number theory, Princeton
University Press, Princeton (1981).
[3] B. J. Green and T. Tao, Primes contain arbitrarily long arithmetic progression, to appear
in Ann. Math.
[4] M. Hamel and I. L
à aba, Arithmetic structures in random sets, Electronic Journal of Combinatorial Number Theory 8 (2008).
[5] Y. Kohayakawa, T. L
à uczak and V. Rödl, Arithmetic progressions of length three in
subsets on a random sets, Acta Arith. 75(1996), 133-163.
[6] K. F. Roth, On certain sets of integers, J. London Math Soc. 28(1953), 245-252.
[7] A. Sárközy, On difference sets of integres III, Acta Math. Sci. Hungar., 31, (1978), 125-149.
[8] E. Szemerédi, On set of integers containing no k elements in arithmetic progressions, Acta
Arith. 27 (1975), 299-345.
[9] T. Tao and T. Ziegler The primes contain arbitrarily long polinomial progressions, to
appear in Acta Math.
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
E-mail address: [email protected]