A Note on Solymosi’s Sum-Product Estimate for Ordered Fields
It is proved that Solymosi’s sum-product estimate max{|A + A|, |A · A|} ≫ |A|4/3/(log |A|)1/3 holds for any finite set A in an ordered field F.
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| author | Xue, Boqing Хуе, Бочінг |
| author_facet | Xue, Boqing Хуе, Бочінг |
| author_sort | Xue, Boqing |
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| description | It is proved that Solymosi’s sum-product estimate max{|A + A|, |A · A|} ≫ |A|4/3/(log |A|)1/3 holds for any finite set A in an ordered field F. |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
Boqing Xue (Shanghai Jiao Tong Univ., China)
A NOTE ON SOLYMOSI’S SUM-PRODUCT ESTIMATE
FOR ORDERED FIELDS*
ПРО ОЦIНКУ ШОЛIМОШI ТИПУ СУМА-ДОБУТОК
ДЛЯ ВПОРЯДКОВАНИХ ПОЛIВ
It is proved that Solymosi’s sum-product estimate max{|A + A|, |A · A|} � |A|4/3/(log |A|)1/3 holds for any finite set
A in an ordered field F.
Доведено, що оцiнка Шолiмошi типу сума-добуток max{|A + A|, |A · A|} � |A|4/3/(log |A|)1/3 справедлива для
будь-якої скiнченної множини A у впорядкованому полi F.
For a set A of a given ring (R,+, ·), define the sum-set and the product-set to be
A+A = {a+ a′ : a, a′ ∈ A},
A ·A = {a · a′ : a, a′ ∈ A}.
When R is a field and 0 /∈ A, we also apply similar definition for A/A.
Since Z and R do not have zero divisors and finite subrings, it is expected that the sum-set and
the product-set can not be relatively small simultaneously. Erdős and Szemerédi [2] conjectured that
for any finite set A ⊆ Z, the estimate (here� and� are Vinogradov notation)
max
{
|A+A|, |A ·A|
}
� |A|2−ε
holds, where ε→ 0 when |A| → ∞. And they proved that
max
{
|A+A|, |A ·A|
}
� |A|1+δ
for some δ > 0. Later Nathanson [6] showed that δ ≥ 1/31 and Ford [3] improved this bound to
δ ≥ 1/15. For finite sets of reals (also correct for finite sets of integers), bounds were given by Elekes
[1] (δ ≥ 1/4), Solymosi [7] (δ ≥ 3/11− ε) and Solymosi [8] (δ ≥ 1/3− ε). The proofs in [1] and
[8] are quite beautiful. Geometry is taken use of in these two papers.
For sum-product estimates for the finite fields and the complex numbers, we refer the reader to
[4, 9, 10].
In this note, Solymosi’s bound is extended to finite sets of any ordered rings. The geometry proof
is transferred to a type of elementary linear algebra.
Definition. An ordered field (or ring) is a field (or ring, respectively) (F,+, ·) with a total order
≤ such that for all a, b and c in F, the following two properties hold:
(i) if a ≤ b, then a+ c ≤ b+ c,
* This work is supported by the National Natural Science Foundation of China (Grant No. 11271249).
c© BOQING XUE, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9 1257
1258 BOQING XUE
(ii) if 0 ≤ a and 0 ≤ b, then 0 ≤ ab.
Examples of ordered fields include Q, R, the field of fractions of R[x] with R an ordered ring,
computable numbers, superreal numbers, hyperreal numbers and so on. One can found details on
Wikipedia.
Theorem. Supose F is an ordered field. Let A ⊆ F be any finite set with sufficiently large
cardinality. Then
|A+A|2|A ·A| � |A|4
log |A|
.
From the theorem one can deduce the follow sum-product estimate.
Corollary. Supose F is an ordered field. Let A ⊆ F be any finite set with sufficiently large
cardinality. Then
max{|A+A|, |A ·A|} � |A|4/3
(log |A|)1/3
.
For a nontrivial ordered ring R, one can find a nonempty set P ⊆ R such that
(i) if a, b ∈ P, then a+ b ∈ P and ab ∈ P,
(ii) for all r ∈ R, exactly one of the following conditions holds:
r ∈ P, r = 0, −r ∈ P.
P is called the positive elements of R and we say r is negative if −r ∈ P. This can be viewed as
an alternative definition of an ordered ring. Now we fix an A ⊆ F and begin to prove the theorem.
Without loss of generality, we suppose that all the elements in A are positive. (Either the set of
positive elements of A or the set of negative ones has cardinality no less than (|A| − 1)/2� |A| and
we can substitute it for original A.) Put Sλ = {(a, b) ∈ A × A : a/b = λ} and rA/A(λ) = |Sλ|. A
trivial bound is rA/A(λ) ≤ |A|. Define the energy by
E×(A) = #{(a, b, c, d) ∈ A4 : ab = cd},
E÷(A) = #{(a, b, c, d) ∈ A4 : a/b = c/d}, 0 /∈ A.
It can be asserted that E×(A) = E÷(A). The energy inequality shows that
|A|4
|A ·A|
≤ E×(A) = E÷(A) =
∑
λ∈A/A
r2A/A(λ).
Let t = dlog |A|/ log 2e, where the notation dxe denote the smallest integer larger than or equal
to x. For 0 ≤ j ≤ t, denote
Mj := {λ ∈ A/A : 2j ≤ rA/A(λ) < 2j+1}, mj := |Mj |.
It follows that
E÷(A) =
t∑
j=0
∑
λ∈Mj
r2A/A(λ) ≤
t∑
j=0
22j+2mj .
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
A NOTE ON SOLYMOSI’S SUM-PRODUCT ESTIMATE FOR ORDERED FIELDS 1259
Hence
|A|4
|A ·A| · log |A|
≤ sup
0≤j≤t
{22j+2mj} := 22J+2mJ . (1)
If mJ = 1, then trivial bound gives
22J+2mJ � 22t � |A|2.
By (1), one has |A · A| · log |A| ≥ |A|2. Combining the trivial bound |A+ A|2 ≥ |A|2, the theorem
follows. Now we suppose that mJ ≥ 2. For µ1, µ2 ∈MJ , we construct a map πµ1,µ2 : Sµ1 × Sµ2 →
→ (A+A)× (A+A):
πµ1,µ2(a1, b1, a2, b2) = (a1 + a2, b1 + b2).
Lemma 1. When µ1 6= µ2, the map πµ1,µ2 is an injection.
Proof. Suppose there exist (a1, b1, a2, b2) and (a′1, b
′
1, a
′
2, b
′
2) in Sµ1 × Sµ2 such that
πµ1,µ2(a1, b1, a2, b2) = πµ1,µ2(a
′
1, b
′
1, a
′
2, b
′
2).
Then we have the following linear equations:
a1 + a2 = a′1 + a′2, (2)
b1 + b2 = b′1 + b′2, (3)
a1/b1 = a′1/b
′
1 = µ1, (4)
a2/b2 = a′2/b
′
2 = µ2. (5)
Substituting (4) and (5) into (2), we obtain
µ1b1 + µ2b2 = µ1b
′
1 + µ2b
′
2.
Then subtract µ1 times (3), we get
(µ2 − µ1)b2 = (µ2 − µ1)b′2.
Since µ1 6= µ2, it appears that b2 = b′2. Now from (2), (4) and (5), we conclude that
(a1, b1, a2, b2) = (a′1, b
′
1, a
′
2, b
′
2).
Lemma 1 is proved.
Lemma 2. If µ1 < µ2 ≤ µ3 < µ4, then
πµ1,µ2(Sµ1 × Sµ2) ∩ πµ3,µ4(Sµ3 × Sµ4) = ∅.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1260 BOQING XUE
Proof. Suppose on the contrary, there exist (a1, b1, a2, b2) ∈ Sµ1 × Sµ2 and (a′1, b
′
1, a
′
2, b
′
2) ∈
∈ Sµ3 × Sµ4 such that
πµ1,µ2(a1, b1, a2, b2) = πµ3,µ4(a
′
1, b
′
1, a
′
2, b
′
2).
Then we have the following linear equations:
a1 + a2 = a′1 + a′2, (6)
b1 + b2 = b′1 + b′2, (7)
a1/b1 = µ1, (8)
a2/b2 = µ2, (9)
a′1/b
′
1 = µ3, (10)
a′2/b
′
2 = µ4. (11)
Substituting (8) – (11) into (6), we obtain
µ1b1 + µ2b2 = µ3b
′
1 + µ4b
′
2.
Combining (7), yields
(µ2 − µ1)b2 = (µ3 − µ1)b′1 + (µ4 − µ1)b′2.
Since µ1 < µ2 ≤ µ3 < µ4, one deduces that
(µ2 − µ1)b2 > (µ2 − µ1)b′1 + (µ2 − µ1)b′2,
i.e., b2 > b′1 + b′2, which is a contradiction to (7) and the fact b1 > 0.
Lemma 2 is proved.
Recall mJ ≥ 2. Write MJ := {λ1, λ2, . . . , λmJ}, where λ1 < λ2 . . . < λmJ . Then
mJ−1⋃
i=1
πλi,λi+1
(
Sλi × Sλi+1
)
⊆ (A+A)× (A+A).
In view of Lemmas 1 and 2, one has∣∣πλi,λi+1
(Sλi × Sλi+1
)
∣∣ = |Sλi | · |Sλi+1
| ≥ 22J
for 1 ≤ i ≤ mJ − 1 and
πλi,λi+1
(
Sλi × Sλi+1
)
∩ πλj ,λj+1
(
Sλj × Sλj+1
)
= ∅
for 1 ≤ i < j ≤ mJ − 1. As a result,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
A NOTE ON SOLYMOSI’S SUM-PRODUCT ESTIMATE FOR ORDERED FIELDS 1261
|A+A|2 ≥
∣∣∣∣∣
mJ−1⋃
i=1
πλi,λh+i
(
Sλi × Sλh+i
)∣∣∣∣∣ =
=
mJ−1∑
i=1
∣∣∣πλi,λmJ−1
(
Sλi × Sλh+i
)∣∣∣ = (mJ − 1) · 22J � mJ · 22J . (12)
Combining (1) and (12), gives
|A+A|2|A ·A| � |A|4
log |A|
.
Remark. For the sum-division estimate, the log |A|-term in the denominator can be eliminated,
using the method from Li [5].
Acknowledgement. The author would like to thank his advisor, Professor Hongze Li, for
continual help and encouragement. He is grateful for Zhen Cui for deep discussions during the
seminars. He is also grateful for Liangpan Li, who gave several talks on this issue during 2010-2011.
1. Elekes Gy. On the number of sums and products // Acta Arith. – 1997. – 81. – P. 365 – 367.
2. Erdős P., Szemerédi E. On sums and products of integers // Stud. Pure Math. – Basel: Birkhäuser, 1983. – P. 213 – 218.
3. Ford K. Sums and products from a finite set of real numbers // Ramanujan J. – 1998. – 2. – P. 59 – 66.
4. Konyagin S. V., Rudnev M. New sum product type estimates // arXiv: math: 1207.6785.
5. Liangpan Li, Jian Shen. A sum-division estimate of reals // Proc. Amer. Math. Soc. – 2010. – 138. – P. 101 – 104.
6. Nathanson M. B. On sums and products of integers // Proc. Amer. Math. Soc. – 1997. – 125. – P. 9 – 16.
7. Solymosi J. On the number of sums and products // Bull. London Math. Soc. – 2005. – 37, № 4. – P. 491 – 494.
8. Solymosi J. Bounding multiplicative energy by the sumset // Adv. Math. – 2009.
9. Rudnev M. An improved sum-product inequality in fields of prime order // arXiv: math: 1011.2738.
10. Rudnev M. On new sum-product type estimates // arXiv: math: 1111.4977.
Received 10.09.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:20:53Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-22162019-12-05T10:26:31Z A Note on Solymosi’s Sum-Product Estimate for Ordered Fields Про оцінку Шолімоші типу сума-добуток для впорядкованих полів Xue, Boqing Хуе, Бочінг It is proved that Solymosi’s sum-product estimate max{|A + A|, |A · A|} ≫ |A|4/3/(log |A|)1/3 holds for any finite set A in an ordered field F. Доведено, що оцінка Шолімоші типу сума-добуток max{|A + A|, |A · A|} ≫ |A|4/3/(log |A|)1/3 справедлива для будь-якої скінченної множини A у впорядкованому полі F. Institute of Mathematics, NAS of Ukraine 2014-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2216 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 9 (2014); 1257–1261 Український математичний журнал; Том 66 № 9 (2014); 1257–1261 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2216/1433 https://umj.imath.kiev.ua/index.php/umj/article/view/2216/1434 Copyright (c) 2014 Xue Boqing |
| spellingShingle | Xue, Boqing Хуе, Бочінг A Note on Solymosi’s Sum-Product Estimate for Ordered Fields |
| title | A Note on Solymosi’s Sum-Product Estimate for Ordered Fields |
| title_alt | Про оцінку Шолімоші типу сума-добуток для впорядкованих полів |
| title_full | A Note on Solymosi’s Sum-Product Estimate for Ordered Fields |
| title_fullStr | A Note on Solymosi’s Sum-Product Estimate for Ordered Fields |
| title_full_unstemmed | A Note on Solymosi’s Sum-Product Estimate for Ordered Fields |
| title_short | A Note on Solymosi’s Sum-Product Estimate for Ordered Fields |
| title_sort | note on solymosi’s sum-product estimate for ordered fields |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2216 |
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