DSpace at VNU: A Szemerédi–Trotter type theorem, sum-product estimates in finite quasifields, and related results

DSpace at VNU: A Szemerédi–Trotter type theorem, sum-product estimates in finite quasifields, and related results tài li...

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estimates in finite quasifields, and related results

Thang Phama,1, Michael Taitb,2, Craig Timmonsc,3,

Le Anh Vinhd,4

aEPFL, Lausanne, Switzerland

bDepartment of Mathematical Sciences, Carnegie Mellon University, United States

cDepartment of Mathematics and Statistics, California State University

Sacramento, United States

d

University of Education, Vietnam National University Hanoi, Viet Nam

a r t i c l e i n f o a b s t r a c t

Article history:

Received 27 June 2015

Available online 6 December 2016

Keywords:

Szemerédi–Trotter theorem

Quasifield

Sum-product estimate

We prove a Szemerédi–Trotter type theorem and a sum-product estimate in the setting of finite quasifields These estimates generalize results of the fourth author, of Garaev, and of Vu We generalize results of Gyarmati and Sárközy on the solvability of the equationsa + b = cd and ab+ 1= cd

over a finite field Other analogous results that are known to hold in finite fields are generalized to finite quasifields.

© 2016 Elsevier Inc All rights reserved.

E-mail addresses:thang.pham@epfl.ch (T Pham), mtait@cmu.edu (M Tait), craig.timmons@csus.edu

(C Timmons), vinhla@vnu.edu.vn (L.A Vinh).

1

The first author was partially supported by Swiss National Science Foundation grants 200020-162884 and 200020-144531.

2 Research supported in part by National Science Foundation Postdoctoral Fellowship 1606350.

3 Research supported in part by Simons Foundation Grant 35419.

4 The fourth author was supported by Vietnam National Foundation for Science and Technology Development grant 101.99-2013.21.

http://dx.doi.org/10.1016/j.jcta.2016.11.003

0097-3165/© 2016 Elsevier Inc All rights reserved.

1 Introduction

Let R be aring andA ⊂ R.Thesumset of A istheset A + A={a + b : a, b ∈ A},

and theproduct set of A istheset A · A={a · b : a, b ∈ A}.A well-studiedproblem in arithmeticcombinatoricsistoprovenon-triviallower boundsonthequantity

max{|A + A|, |A · A|}

undersuitablehypothesisonR and A.OneofthefirstresultsofthistypeisduetoErdős and Szemerédi [8] They proved thatif R = Z and A is finite,then there are positive constants c and ,bothindependentofA, suchthat

max{|A + A|, |A · A|} ≥ c|A| 1+

Thisimprovesthetriviallowerboundofmax{|A +A |, |A ·A|} ≥ |A|.ErdősandSzemerédi conjecturedthatthe correctexponentis 2− o(1) where o(1) → 0 as |A| → ∞.Despite

asignificant amountof researchonthisproblem,this conjectureisstillopen Forsome time the best known exponent was 4/3 − o(1) due to Solymosi [22] (see also [17] for similar results)whoprovedthatforany finitesetA ⊂ R,

max{|A + A|, |A · A|} ≥ |A| 4/3

2(log|A|) 1/3

Very recently,KonyaginandShkredov[18] announcedanimprovementoftheexponent

to 4/3 + c − o(1) for anyc < 205981

AnothercasethathasreceivedattentioniswhenR is afinite field.Letp beaprime and let A ⊂ Z p Bourgain, Katz, and Tao [1] proved that if p δ < |A| < p1−δ where

0< δ < 1/2,then

for somepositive constants c and  dependingonly onδ. Hart, Iosevich, andSolymosi [14] obtainedboundsthatgiveanexplicitdependenceof on δ. Letq beapowerof an oddprime, Fq be thefinite fieldwith q elements, andA ⊂ F q In[14],itis shownthat

if|A + A | = m and |A · A| = n,then

|A|3≤ cm2n |A|

where c is somepositiveconstant.Inequality (2) impliesa non-trivial sum-product es-timate when q 1/2  |A|  q. We write f  g if f = o(g). Using a graph theoretic approach, the fourthauthor [26] and Vu [29] improved(2) and as a result,obtained a bettersum-productestimate

Theorem 1.1 ( [26] ) Let q be a power of an odd prime If A ⊂ F q , |A + A | = m, and

|A · A| = n, then

|A|2≤ mn |A|

1/2 √ mn.

Corollary1.2( [26] ) If q is a power of an odd prime and A ⊂ F q , then there is a positive constant c such that the following hold If q 1/2  |A| < q 2/3 , then

max{|A + A|, |A · A|} ≥ c |A|2

q 1/2

If q 2/3 ≤ |A|  q, then

max{|A + A|, |A · A|} ≥ c(q|A|) 1/2

Inthecasethatq isaprime,Corollary 1.2wasprovedbyGaraev[9]usingexponential sumsandRudnevgaveanestimateforsmallsets [19].Cilleruelo[3]alsoprovedrelated resultsusingdenseSidonsetsinfinitegroupsinvolvingFqandF∗

q.Inparticular,versions

of Theorem 1.3 and (3) (see below) are proved in [3], as well as several other results concerningequationsinFqand sum-productestimates

Theorem 1.1wasprovedusingthefollowingSzemerédi–TrottertypetheoreminFq Theorem1.3( [26] ) Let q be a power of an odd prime If P is a set of points and L is a set of lines in F2, then

|{(p, l) ∈ P × L : p ∈ l}| ≤ |P ||L|

1/2

|P ||L|.

We remark that aSzemerédi–Trotter type theorem in Zp was obtained in [1]using thesum-productestimate(1)

In this paper, we generalize Theorem 1.1, Corollary 1.2, and Theorem 1.3 to finite quasifields.Werecallthedefinitionofaquasifieldnow:AsetL withabinaryoperation·

iscalledaloop if

1 theequationa · x = b hasauniquesolutioninx foreverya, b ∈ L,

2 theequationy · a = b hasauniquesolutioniny foreverya, b ∈ L,and

3 thereisanelemente ∈ L suchthate · x = x · e = x forallx ∈ L.

A (left) quasifield Q is aset with two binary operations+ and · such that(Q,+) is a groupwith additive identity 0, (Q ∗ , ·) is aloop where Q ∗ = Q \{0}, and thefollowing threeconditionshold:

1 a · (b + c) = a · b + a · c foralla, b, c ∈ Q,

2 0· x= 0 forallx ∈ Q,and

3 theequationa · x = b · x + c hasexactlyonesolutionforeverya, b, c ∈ Q with a

Any finitefieldisaquasifield.There aremanyexamplesofquasifields whicharenot fields;seeforexample,Chapter5of[6]orChapter9of[16].Quasifieldsappearextensively

inthetheoryofprojectiveplanes.Wenotethatinparticular,inaquasifieldmultiplication neednotbecommutativenorassociative.Throughoutthepaperwemustbecarefulabout which sidemultiplication takesplace on,and be warythatmultiplicativeinversesneed notexistonbothsides.Nonassociativityofmultiplicationisabiggerproblem.Previous researchonsum-productestimatesrequiresassociativityofmultiplicationfortoolssuch

asPlünnecke’sinequality(seeforexample,[23]forthemostgeneralknownsum-product theorem,theproofofwhichusesassociativityofmultiplicationthroughout)

Theorem 1.4 Let Q be a finite quasifield with q elements If A ⊂ Q\{0}, |A + A | = m,

and |A · A| = n, then

|A|2≤ mn |A|

1/2 √ mn.

Theorem 1.4givesthefollowingsum-productestimate

Corollary 1.5.Let Q be a finite quasifield with q elements and A ⊂ Q\{0} There is a positive constant c such that the following hold.

If q 1/2  |A| < q 2/3 , then

max{|A + A|, |A · A|} ≥ c |A|2

q 1/2

If q 2/3 ≤ |A|  q, then

max{|A + A|, |A · A|} ≥ c(q|A|) 1/2

FromCorollary 1.5weconcludethatanyalgebraicobjectthatisrichenoughto coordi-natizeaprojectiveplanemustsatisfyanon-trivialsum-productestimate.Following[26],

weproveaSzemerédi–TrottertypetheoremandthenuseittodeduceTheorem 1.4.We note thattheconnectionbetweenarithmeticcombinatoricsandincidencegeometrywas studied in a general form in [10] We also note that many authors havestudied more general incidence theorems and their relationshipto arithmeticcombinatorics (cf [13, 15,4,5])

Theorem 1.6 Let Q be a finite quasifield with q elements If P is a set of points and L

is a set of lines in Q2, then

|{(p, l) ∈ P × L : p ∈ l}| ≤ |P ||L|

1/2

|P ||L|.

AnotherconsequenceofTheorem 1.6 isthefollowingcorollary.

Corollary 1.7.If Q is a finite quasifield with q elements and A ⊂ Q, then there is a positive constant c such that

|A · (A + A)| ≥ c min



q, |A|3

q



Further, if |A| q 2/3 , then one may take c= 1+ o(1).

ThenextresultgeneralizesTheorem 1.1from[28]

Theorem1.8 Let Q be a finite quasifield with q elements If A, B, C ⊂ Q, then

|A + B · C| ≥ q − |A||B||C| + q q3 2.

WenotethatCorollary 1.7appliestoelementsoftheforma · b + a · c where a, b, c ∈ A

and Theorem 1.8 applies to elements of the form a + b · c where a ∈ A, b ∈ B, and

c ∈ C.Theorem 1.8does notuseourSzemerédi–TrotterTheorem,andits proofallows for the more general result of taking three distinct sets, whereas Corollary 1.7 is not

as flexible,butgivesabetterestimatewhen|A| is betweenq 1/3 andq 2/3.Thespirit of these tworesultsis similar,though itisnotclear inthe settingof aquasifield thatthe sets A · (A + A) and A + A · A shouldnecessarily behave thesame way (it isalso not clearthattheyshouldn’t)

Ourmethodsinprovingtheaboveresultscanbeusedtogeneralizetheorems concern-ingthesolvabilityofequationsoverfinitefields.Letp beaprimeandletA, B, C, D ⊂ Z p Sárközy[20] provedthatifN (A, B, C, D) isthenumberofsolutionstoa + b = cd with (a, b, c, d) ∈ A × B × C × D,then



N(A,B,C,D) − |A||B||C||D| p 

 ≤ p 1/2

Inparticular,if|A||B||C||D| > p3,thenthereisan(a, b, c, d) ∈ A × B × C × D suchthat

a + b = cd.This is best possible up to a constantfactor (see [20]) It wasgeneralized

to finite fields of odd prime power order by Gyarmati and Sárközy [11], and then by thefourthauthor[25] to systemsof equationsover Fq Here wegeneralizethe resultof Gyarmatiand Sárközytofinite quasifields

Theorem 1.9.Let Q be a finite quasifield with q elements and let A, B, C, D ⊂ Q If

γ ∈ Q and N γ (A, B, C, D) is the number of solutions to a + b + γ = c · d with a ∈ A,

b ∈ B, c ∈ C, and d ∈ D, then



N γ (A, B, C, D) − (q + 1) |A||B||C||D|

q2+ q + 1



 ≤ q 1/2

|A||B||C||D|.

Theorem 1.9impliesthefollowingCorollarywhichgeneralizesCorollary 3.5in[27].

Corollary 1.10.If Q is a finite quasifield with q elements and A, B, C, D ⊂ Q with

|A||B||C||D| > q3, then

Q = A + B + C · D.

WealsoproveahigherdimensionalversionofTheorem 1.9

Theorem 1.11.Let d ≥ 1 be an integer If Q is a finite quasifield with q elements and

A ⊂ Q with |A| ≥ 2q 2d+2 d+2 , then

Q = A + A + A · A + · · · + A · A

d terms

.

AnotherproblemconsideredbySárközywasthesolvabilityoftheequationab+1= cd

overZp.Sárközy[21]provedaresultinZp whichwaslatergeneralizedtothefinitefield setting in[11]

Theorem1.12(Gyarmati, Sárközy) Let q be a power of a prime and A, B, C, D ⊂ F q If

N (A, B, C, D) is the number of solutions to ab+ 1= cd with a ∈ A, b ∈ B, c ∈ C, and

d ∈ D, then



N(A,B,C,D) − |A||B||C||D| q 

 ≤ 8q 1/2(|A||B||C||D|) 1/2 + 4q2.

Ourgeneralizationto quasifieldsisasfollows

Theorem1.13.Let Q be a finite quasifield with q elements and kernel K Let γ ∈ Q\{0}, and A, B, C, D ⊂ Q If N γ (A, B, C, D) is the number of solutions to a · b + c · d = γ, then



N γ (A, B, C, D) − |A||B||C||D|

q



 ≤ q |A||B||C||D| |K| − 1 1/2

Corollary1.14.Let Q be a quasifield with q elements whose kernel is K If A, B, C, D ⊂ Q and |A||B||C||D| > q4(|K| − 1) −1 , then

Q \{0} ⊂ A · B + C · D.

ByappropriatelymodifyingtheargumentusedtoproveTheorem 1.13, wecanprove

ahigherdimensionalversion

Theorem 1.15.Let Q be a finite quasifield with q elements whose kernel is K If A ⊂ Q and |A| > q1+1d(|K| − 1) −1/2d , then

Q \{0} ⊂ A · A + · · · + A · A 

d terms

.

IfQ isafinitefield,then|K| = q,and theboundsofTheorems 1.13 and 1.15match theboundsobtainedbyHart andIosevichin[12]

Finally,wenotethatourtheoremsareprovedusingspectraltechniques.Intheproofs,

if the size of the set is small, the error term from spectral estimates will dominate Therefore,theresultspresentedareonlynontrivialifthesize ofthesetislargeenough Sum-productestimatesforsmallsetshavebeengiven(forexamplein[1,17,23]).Wealso notethatitisnothardtoshowthatonemayfindasetA ineitherafield,generalring,

orquasifield,whereboth |A + A | and |A · A| areoforder|A|2

Therestofthepaperisorganizedasfollows.InSection2wecollectsomepreliminary results.Section3containstheproofofTheorem 1.4,1.6,and1.9,aswellasCorollary 1.5, 1.7,and 1.10 Section4containstheproof ofTheorem 1.8 and1.11.Section5contains theproofofTheorem 1.13 and1.15

2 Preliminaries

Webeginthissectionbygivingsomepreliminaryresultsonquasifields.LetQ denote

afinitequasifield.Weuse1todenotetheidentityintheloop(Q ∗ , ·).Itisaconsequence

of the definition that (Q,+) must be an abelian group One also has x · 0 = 0 and

x · (−y) = −(x · y) for all x, y ∈ Q (see [16], Lemma 7.1) For more on quasifields, see Chapter 9 of [16] A (right) quasifield is required to satisfy the right distributive lawinstead ofthe left distributivelaw Thekernel K of aquasifield Q is theset of all elementsk ∈ Q thatsatisfy

1 (x + y) · k = x · k + y · k forallx, y ∈ Q,and

2 (x · y) · k = x · (y · k) forallx, y ∈ Q.

Notethat(K,+) isanabelian subgroupof(Q,+) and(K ∗ , ·) isagroup

Lemma2.1 If a ∈ Q and λ ∈ K, then −(a · λ)= (−a) · λ.

Proof First we show that a · (−1) = −a. Indeed, a · (1+ (−1)) = a · 0 = 0 and so

a + a · (−1)= 0.Weconcludethat−a = a · (−1).Ifλ ∈ K,then

−(a · λ) = a · (−λ) = a · (0 − λ) = a · ((0 − 1) · λ)

= (a · (0 − 1)) · λ = (0 + a · (−1)) · λ = (−a) · λ 2

Fortherestofthissection,weassumethatQ isafinitequasifieldwith|Q| = q.Wecan constructaprojective planeΠ= (P, L, I) that iscoordinatizedbyQ. Here I ⊂ P × L

is the set of incidences between points and lines If p ∈ P and ∈ L, we write p Il to

denote that (p, l) ∈ I,i.e that p is incidentwith l. Wewill follow the notationof [16] and referthereadertoChapter 5of[16] formoredetails.Let∞ beasymbolnotinQ.

Thepointsof Π aredefinedas

P = {(x, y) : x, y ∈ Q} ∪ {(x) : x ∈ Q} ∪ {(∞)}.

Thelines ofΠ aredefinedas

L = {[m, k] : m, k ∈ Q} ∪ {[m] : m ∈ Q} ∪ {[∞]}.

TheincidencerelationI is definedaccordingtothefollowing rules:

1 (x, y) I[m, k] ifandonlyifm · x + y = k,

2 (x, y) I[k] ifand onlyifx = k,

3 (x) I[m, k] ifandonlyifx = m,

4 (x) I[∞] forallx ∈ Q, (∞)I[k] forallk ∈ Q, and(∞)I[∞].

Since |Q| = q,theplaneΠ hasorderq.

NextweassociateagraphtotheplaneΠ.LetG(Π) bethebipartitegraphwithparts

P and L where p ∈ P is adjacent to ∈ L if and onlyif p Il in Π Thefirst lemma is known(see[2],page432)

Lemma 2.2 The graph G(Π) has eigenvalues q + 1 and −(q + 1), each with multiplicity one All other eigenvalues of G(Π) are ±q 1/2

Thenextlemmaisabipartiteversionofthewell-knownExpander MixingLemma Lemma2.3(Bipartite Expander Mixing Lemma) Let G be a d-regular bipartite graph on

2n vertices with parts X and Y Let M be the adjacency matrix of G Let d = λ1≥ λ2≥

· · · ≥ λ 2n =−d be the eigenvalues of M and define λ= maxi=1,2n |λ i | Let S ⊂ X and

T ⊂ Y , and let e(S, T ) denote the number of edges with one endpoint in S and the other

in T Then



e(S,T) − d |S||T | n  ≤ λ|S||T |.

Proof Assume that the columns of M have been ordered so that the columns cor-responding to thevertices of X come beforethecolumns corresponding to thevertices

of Y ForasubsetB ⊂ V (G),letχ BbethecharacteristicvectorforB.Let{x1, , x 2n }

be anorthonormalset ofeigenvectors forM Note thatsinceG isad-regularbipartite graph,wehave

x1= √1

x 2n= √1

Now χ T

S M χ T = e(S, T ).Expanding χ S and χ T as linear combinations of eigenvectors yields

e(S, T ) =

2n

i=1

S , x i x i

T

M

2n

i=1

T , x i x i



=

2n

i=1

S , x i T , x i λ i

Nowby(4)and(5), S , x1= S , x 2n = √1

2n |S| and T , x1= T , x 2n =√1

2n |T |.

Sinceλ1=−λ 2n = d,wehave



e(S,T) − 2d |S||T |

2n



 =2n −1

i=2

S , x i T , x i λ i







≤ λ

2n −1

i=2

S , x i T , x i |

≤ λ

2n −1

i=2

S , x i 2

1/2 2n −1

i=2

T , x i 2

1/2

(by Cauchy–Schwarz)

FinallybythePythagoreanTheorem,

2n −1

i=2

S , x i 2=|S| −2|S|2

2n < |S|

and

2n −1

i=2

T , x i 2=|T | −2|T |2

2n < |T | 2

CombiningLemmas 2.2 and2.3givesthenextlemma

Lemma2.4 For any S ⊂ P and T ⊂ L,



e(S,T) − (q + 1) |S||T |

q2+ q + 1



 ≤ q 1/2

|S||T | where e(S, T ) is the number of edges in G(Π) with one endpoint in S and the other in T

Wenow stateprecisely whatwemeanbyalineinQ2.

Definition 2.5.Givena, b ∈ Q,aline inQ2 isaset oftheform

{(x, y) ∈ Q2: y = b · x + a} or {(a, y) : y ∈ Q}.

When multiplication is commutative, b · x + a = x · b + a. In general, the binary operation· neednotbecommutativeandsowewriteourlineswiththeslopeontheleft The nextlemma is due to Elekes [7](see also [24], page315) Inworkingina (left) quasifield,whichisnotrequiredto satisfytherightdistributivelaw, somecaremustbe takenwithalgebraic manipulations

Lemma 2.6 Let A ⊂ Q ∗ . There is a set P of |A + A ||A · A| points and a set L of |A|2

lines in Q2 such that there are at least |A|3 incidences between P and L.

Proof Let P = (A + A) × (A · A) and

l(a, b) = {(x, y) ∈ Q2: y = b · x − b · a}.

Let L={l(a, b) : a, b ∈ A}. Thestatement that|P |=|A + A ||A · A| is clearfrom the definitionofP Supposel(a, b) and l(c, d) areelementsofL and l(a, b) = l(c, d).Weclaim that(a, b) = (c, d).Inaquasifield,onehasx · 0= 0 foreveryx,andx · (−y)=−(x · y)

for everyx and y ([16],Lemma7.1) Theline l(a, b) contains thepoints (0, −b · a) and

(1, b − b · a). Furthermore,these are theunique pointsinl(a, b) withfirst coordinate0 and 1,respectively.Similarly,thelinel(c, d) containsthepoints(0, −d ·c) and (1, d −d ·c).

Sincel(a, b) = l(c, d),wemusthavethat−b ·a=−d ·c and b −b ·a = d −d ·c.Thus,b = d

andsob · a = b · c.Wecanrewritethisequationasb · a − b · c= 0.Since−x · y = x · (−y)

andQ satisfiestheleftdistributivelaw,wehaveb ·(a −c)= 0.Ifa = c,then(a, b) = (c, d)

and we are done Assumethata that a − c Then we must haveb = 0 for

if b then the product b · (a − c) would be contained in Q ∗ as multiplication is a binary operation onQ ∗.Since A ⊂ Q ∗, wehaveb It mustbe thecase thata = c.

Weconcludethateachpair(a, b) ∈ A2 determinesauniqueline inL and so|L|=|A|2 Consideratriple(a, b, c) ∈ A3.Thepoint(a + c, b · c) belongstoP andisincidentto

l(a, b) ∈ L since

b · (a + c) − b · a = b · a + b · c − b · a = b · c.

EachtripleinA3generatesanincidenceandsothereareatleast|A|3incidencesbetween

P and L. 2

Wenow stateprecisely whatwemeanbyalineinQ2.

Definition 2.5.Givena, b ∈ Q,aline in< i>Q2 isaset oftheform... · a + b · c − b · a = b · c.

EachtripleinA< /i>3generatesanincidenceandsothereareatleast |A| 3incidencesbetween

P and L.