NEW UPPER BOUND FOR SUMS OF DILATES
ALBERT BUSH AND YI ZHAO
Abstract. For λ ∈ Z, let λ · A = {λa : a ∈ A}. We prove that if |A + A| ≤ K|A| and
|λi | ≤ 2r then
2
|λ1 · A + . . . + λh · A| ≤ K O((r+h) / ln(r+h)) |A|.
When r and h are sufficiently large and comparable to each other, this improves upon a
result of Bukh who shows that
|λ1 · A + . . . + λh · A| ≤ K O(rh) |A|.
Our main technique is to combine Bukh’s idea of considering the binary expansion of λi
with a result of Chung, Erdős, and Spencer on biclique decompositions.
1. Introduction
Let A and B be nonempty subsets of an abelian group, and define the sumset of A and
B and the h-fold sumset of A as
A + B := {a + b : a ∈ A, b ∈ B} and hA := {a1 + . . . + ah : ai ∈ A},
respectively. When the set A is implicitly understood, we will reserve the letter K to denote
the doubling constant of A; that is, K := |A+A|/|A|. A classical result of Plünnecke bounds
the cardinality of hA in terms of K and |A|.
Theorem 1 (Plünnecke’s inequality [5]). For any set A and for any nonnegative integers `
and m, if |A + A| = K|A|, then
|`A − mA| ≤ K `+m |A|.
See the survey of Ruzsa [6] for variations, generalizations, and a graph theoretic proof of
Theorem 1; see Petridis [4] for a new inductive proof.
Given λ ∈ Z, define a dilate of A as
λ · A := {λa : a ∈ A}.
Suppose λ1 , . . . , λh are nonzero integers. Since λi · A ⊆ λi A, one can apply Theorem 1 to
conclude that
|λ1 · A + . . . + λh · A| ≤ K
P
i
|λi |
|A|.
Bukh [1] significantly improved this by considering the binary expansion of λi and using
Ruzsa’s covering lemma and triangle inequality.
Theorem 2 (Bukh [1]). For any set A, if λ1 , . . . , λh ∈ Z \ {0} and |A + A| = K|A|, then
|λ1 · A + . . . + λh · A| ≤ K 7+12
Ph
i=1
log2 (1+|λi |)
Date: July 16, 2016.
The second author is partially supported by NSF grant DMS-1400073.
1
|A|.
2
ALBERT BUSH AND YI ZHAO
If |λi | ≤ 2r for all i, then Theorem 2 yields that
|λ1 · A + . . . + λh · A| ≤ K O(rh) |A|.
(1)
In this paper we prove a bound that improves (1) when r and h are sufficiently large and
comparable to each other.
Theorem 3. For any ε > 0, there exists an n0 = n0 (ε) such that for any r, h ∈ Z+ with
r + h ≥ n0 the following holds: for any set A and for any nonzero integers λ1 , . . . , λh such
that |λi | ≤ 2r , if |A + A| = K|A|, then
(r+h)2
|λ1 · A + . . . + λh · A| ≤ K (1+ε) ln(r+h) |A|.
The proof of Theorem 3 is short and relies on Theorem 2 as well as a result of Chung, Erdős,
and Spencer [2] on biclique decompositions. The key idea is to connect Bukh’s technique of
considering the binary expansion of λi to a graph decomposition problem that allows us to
efficiently group certain powers of 2.
We remark here that in all of the above theorems, the condition |A + A| = K|A| can be
replaced with |A − A| = K|A| with no change to the conclusion. It is likely that Theorem 3
is not best possible – we discuss this in the last section.
2. Basic Tools
We need the following variant of Ruzsa’s triangle inequality that we will refer to by the
same name.
Theorem 4 (Ruzsa’s triangle inequality [6]). For any sets X, Y, and Z,
|X + Y | ≤
|X + Z||Z + Y |
.
|Z|
A useful corollary of Ruzsa’s triangle inequality is the following.
Corollary 5. For any sets A and B, if p1 and p2 are nonnegative integers and |A + A| ≤
K|A|, then
|B + p1 A − p2 A| ≤ K p1 +p2 +1 |B + A|.
Proof. Apply Ruzsa’s triangle inequality (Theorem 4) with X = B, Y = p1 A − p2 A, and
Z = A, then apply Plünnecke’s inequality (Theorem 1).
We can use Corollary 5 to prove the following proposition which we will use in the proof
of Theorem 3.
Proposition 6. For any positive integer q, if k1 , `1 , . . . , kq , `q , K are nonnegative integers
and A1 , . . . , Aq , C are sets such that |Ai + Ai | ≤ K|Ai |, then
(2)
|C + k1 A1 − `1 A1 + . . . + kq Aq − `q Aq | ≤ |C + A1 + . . . + Aq | · K q+
Pq
i=1 (ki +`i )
In particular,
(3)
Pq
|k1 A1 − `1 A1 + . . . + kq Aq − `q Aq | ≤ |A1 + . . . + Aq | · K q+
i=1 (ki +`i )
.
.
NEW UPPER BOUND FOR SUMS OF DILATES
3
Proof. (3) follows from (2) by taking C to be a set with a single element so it suffices to prove
(2). We proceed by induction on q. The case q = 1 follows from Corollary 5 immediately.
When q > 1, suppose the statement holds for any positive integer less than q. Applying
Corollary 5 with B = C + k1 A1 − `1 A1 + . . . + kq−1 Aq−1 − `q−1 Aq−1 and A = Aq , we obtain
that
|C + k1 A1 − `1 A1 + . . . + kq Aq − `q Aq |
(4)
≤ K kq +`q +1 |C + k1 A1 − `1 A1 + . . . + kq−1 Aq−1 − `q−1 Aq−1 + Aq |.
Now, let C 0 = C + Aq and apply the induction hypothesis to conclude that
|C 0 + k1 A1 − `1 A1 + . . . + kq−1 Aq−1 − `q−1 Aq−1 |
(5)
Pq−1
≤ |C 0 + A1 + . . . + Aq−1 | · K q−1+
i=1 (ki +`i )
Combining (4) with (5) gives the desired inequality:
|C + k1 A1 − `1 A1 + . . . + kq Aq − `q Aq | ≤ |C + A1 + . . . + Aq | · K q+
Pq
i=1 (ki +`i )
.
3. Proof of Theorem 3
Given λ1 , . . . , λh ∈ Z \ {0}, we define
r := maxblog2 |λi |c
(6)
i
and write the binary expansion of λi as
(7)
λi = i
r
X
λi,j 2j , where λi,j ∈ {0, 1} and i ∈ {−1, 1}.
j=0
Bukh’s proof of Theorem 2 actually gives the following stronger statement which we will use
in the proof of Theorem 3.
Theorem 7 ([1]). If λ1 , . . . , λh ∈ Z \ {0} and |A + A| = K|A|, then
|λ1 · A + . . . + λh · A| ≤ K 7+10r+2
P P
i
j
λi,j
|A|.
In the proof of Theorem 2, the first step is to observe that
r
r
X
X
j
λ1 · A + . . . + λh · A ⊆
(λ1,j 2 ) · A + . . . +
(λh,j 2j ) · A.
j=0
j=0
In our proof, we also consider the binary expansion of λi , but we do the above step more
efficiently by first grouping together λi that have shared binary digits. In order to do this
systematically, we view the problem as a graph theoretic problem and apply the following
result on biclique decompositions.
Theorem 8 (Chung, Erdős, Spencer [2]). Given ε0 > 0, there exists n0 = n0 (ε0 ) such that
every graph G on n ≥ n0 vertices can be decomposed into edge-disjoint complete bipartite
subgraphs H1 , . . . , Hq such that E(G) = ∪i E(Hi ) and
q
X
i=1
|V (Hi )| ≤ (1 + ε0 )
n2
.
2 ln n
4
ALBERT BUSH AND YI ZHAO
Proof of Theorem 3. Given ε > 0, let r + h be sufficiently large. Given nonzero integers
λ1 , . . . , λh , define r and λi,j as in (6) and (7). We define a bipartite graph G = (X, Y, E)
as follows: let X = {λ1 , . . . , λh }, Y = {20 , . . . , 2r }, and E = {(λi , 2j ) : λi,j = 1}. In other
words, λi is connected to the powers of 2 that are present in its binary expansion.
We apply Theorem 8 to G with ε0 = ε/3 and obtain a biclique decomposition H =
{H1 , . . . , Hq } of G. That is, H is a set of edge-disjoint complete bipartite graphs whose
union covers the edge set of G. Furthermore, assume Hi := (Xi , Yi , Ei ) where Xi ⊆ X,
Yi ⊆ Y . We have Ei = {(u, v) : u ∈ Xi , v ∈ Yi } and
q
X
ε (r + h + 1)2
ε (r + h)2
|Xi | + |Yi | ≤ 1 +
(8)
≤ 1+
3 2 ln(r + h + 1)
2 2 ln(r + h)
i=1
as r + h is sufficiently large.
Now, we connect this biclique decomposition to the sum of dilates λ1 ·A+. . .+λh ·A. Since
the elements of X and Y are integers, we can perform arithmetic operations with them. For
j = 1, . . . , q, let
X
γj :=
y,
y∈Yj
and since H is a biclique decomposition, for i = 1, . . . , h, we have
X
λi = i
γj .
j:λi ∈Xj
Applying the above to each λi along with the fact that A + (α + β) · A ⊆ A + α · A + β · A
results in
X
X
(9)
λ1 · A + . . . + λh · A ⊆ 1
(γj · A) + . . . + h
(γj · A).
j:λ1 ∈Xj
j:λh ∈Xj
Let kj := |{λi ∈ Xj : λi > 0}|, `j := |{λi ∈ Xj : λi < 0}|, and note that kj + `j = |Xj |. By
regrouping the terms in (9), we have
X
X
γj · A
γj · A + . . . + h
1
j:λh ∈Xj
j:λ1 ∈Xj
= k1 (γ1 · A) − `1 (γ1 · A) + . . . + kq (γq · A) − `q (γq · A).
Since |γj · A + γj · A| = |A + A| ≤ K|A| = K|γj · A|, we can apply Proposition 6 to conclude
that
|k1 (γ1 · A) − `1 (γ1 · A) + . . . + kq (γq · A) − `q (γq · A)|
(10)
Pq
≤ |γ1 · A + . . . + γq · A| · K q+
i=1
ki +`i
≤ |γ1 · A + . . . + γq · A| · K 2
Pq
i=1
|Xi |
.
For 1 ≤ i ≤ q and 0 ≤ j ≤ r, let γi,j = 1 if 2j is in the binary expansion of γi and 0 otherwise.
Observe that
r
X
maxblog2 γj c ≤ maxblog2 |λi |c = r and
γi,j = |Yi |.
j
i
j=0
Hence, by Theorem 7,
(11)
|γ1 · A + . . . + γq · A| ≤ K 7+10r+2
Pq
i=1
Pr
j=0
γi,j
|A| = K 7+10r+2
Pq
i=1
|Yi |
|A|.
NEW UPPER BOUND FOR SUMS OF DILATES
5
Combining (10) and (11) with (8) results in
|λ1 · A + . . . + λh · A| ≤ K 7+10r+2
Pq
i=1 (|Xi |+|Yi |)
(r+h)2
|A| ≤ K (1+ε) ln(r+h) |A|,
where the second inequality holds also because r + h is sufficiently large.
4. Concluding Remarks
• Chung, Erdős, and Spencer [2] also proved that there exist graphs G such that for any
biclique decomposition H1 , . . . , Hq ,
q
X
i=1
n2
.
|V (Hi )| >
2e ln n
Moreover, one can modify their construction to ensure that G is in fact bipartite. Hence,
one cannot expect to improve Theorem 3 by improving the corresponding result for clique
partitions.
• If each of λ1 , . . . , λr has O(1) digits in its binary expansion, then Theorem 7 yields that
|λ1 · A + . . . + λr · A| ≤ K O(r) |A|. Bukh asked if this bounds holds whenever λi ≤ 2r :
Question 9 (Bukh [1]). For any set A and for any λ1 , . . . , λr ∈ Z \ {0}, if |A + A| = K|A|
and 0 < λi ≤ 2r , then
|λ1 · A + . . . + λr · A| ≤ K O(r) |A|.
In light of Question 9, one can view Theorem 3 as providing modest progress by proving
a subquadratic bound of quality O(r2 / ln r) whereas Theorem 2 shows that the exponent
is O(r2 ).
• Generalized arithmetic progressions give supporting evidence for Question 9. A generalized
arithmetic progression P is a set of the form
P := {d + x1 d1 + . . . + xk dk : 0 ≤ xi < Li }.
Moreover, P is said to be proper if |P | = L1 · . . . · Lk . One can calculate that if P is proper,
then
|P + P | ≤ (2Li − 1)k ≤ 2k |P | =: K|P |.
Additionally, one can calculate that for any λ1 , . . . , λh ∈ Z+ , if λi ≤ 2r then
|λ1 · P + . . . + λh · P | ≤ (λ1 + . . . + λh )k |P | = 2k log2 (λ1 +...+λh ) |P | = K r+log2 h |P |.
Freiman’s theorem [3] states that, roughly speaking, sets with small doubling are contained in generalized arithmetic progressions with bounded dimension. Using this line of
reasoning, Schoen and Shkredov [7] proved that
|λ1 · A + . . . + λh · A| ≤ eO((log
8
K)(h+log
P
i
|λi |))
|A|.
This naturally leads us to ask a more precise version of Question 9.
Question 10. If |A + A| = K|A|, then is
|λ1 · A + . . . + λh · A| ≤ K O(h+log
P
i
|λi |)
|A|?
Acknowledgment
We thank Boris Bukh and Giorgis Petridis for reading an earlier version of this paper and
giving valuable comments.
6
ALBERT BUSH AND YI ZHAO
References
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[2] Fan Chung, Paul Erdős, and Joel Spencer. On the decomposition of graphs into complete bipartite
subgraphs. In Studies in Pure Mathematics, pages 95–101. Springer, 1983.
[3] Gregory A. Freı̆man. Foundations of a structural theory of set addition. American Mathematical Society,
Providence, R. I., 1973. Translated from the Russian, Translations of Mathematical Monographs, Vol 37.
[4] Giorgis Petridis. New proofs of Plünnecke-type estimates for product sets in groups. Combinatorica,
32(6):721–733, 2012.
[5] Helmut Plünnecke. Eine zahlentheoretische anwendung der graphtheorie. J. Reine Angew. Math.,
243:171–183, 1970.
[6] Imre Z Ruzsa. Sumsets and structure. In Combinatorial Number Theory and Additive Group Theory,
pages 87–210, 2009.
[7] Tomasz Schoen and Ilya D. Shkredov. Additive dimension and a theorem of Sanders. J. Aust. Math. Soc.,
100(1):124–144, 2016.
Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303
E-mail address, Albert Bush: [email protected]
E-mail address, Yi Zhao: [email protected]